![]() |
Metamath
Proof Explorer Theorem List (p. 35 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30438) |
![]() (30439-31961) |
![]() (31962-47939) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | reubida 3401 | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rmobidvaOLD 3402* | Obsolete version of rmobidv 3391 as of 23-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | cbvrmow 3403* | Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3423 with a disjoint variable condition, which does not require ax-10 2135, ax-13 2369. (Contributed by NM, 16-Jun-2017.) Avoid ax-10 2135, ax-13 2369. (Revised by Gino Giotto, 23-May-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreuw 3404* | Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3422 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2135. (Revised by Wolf Lammen, 10-Dec-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfrmo1 3405 | The setvar 𝑥 is not free in ∃*𝑥 ∈ 𝐴𝜑. (Contributed by NM, 16-Jun-2017.) |
⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝜑 | ||
Theorem | nfreu1 3406 | The setvar 𝑥 is not free in ∃!𝑥 ∈ 𝐴𝜑. (Contributed by NM, 19-Mar-1997.) |
⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜑 | ||
Theorem | nfrmow 3407* | Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3428 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2114, ax-ext 2701. (Revised by Wolf Lammen, 21-Nov-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfreuw 3408* | Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3429 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2114, ax-ext 2701. (Revised by Wolf Lammen, 21-Nov-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 | ||
Theorem | cbvrmowOLD 3409* | Obsolete version of cbvrmow 3403 as of 23-May-2024. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreuwOLD 3410* | Obsolete version of cbvreuw 3404 as of 10-Dec-2024. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreuvwOLD 3411* | Obsolete version of cbvreuvw 3398 as of 30-Sep-2024. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | rmoeq1 3412* | Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2106. (Revised by Wolf Lammen, 12-Mar-2025.) |
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | reueq1 3413* | Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2106. (Revised by Wolf Lammen, 12-Mar-2025.) |
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rmoeq1OLD 3414* | Obsolete version of rmoeq1 3412 as of 12-Mar-2025. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | reueq1OLD 3415* | Obsolete version of reueq1 3413 as of 12-Mar-2025. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | rmoeqd 3416* | Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | reueqd 3417* | Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | rmoeq1f 3418 | Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | reueq1f 3419 | Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | nfreuwOLD 3420* | Obsolete version of nfreuw 3408 as of 21-Nov-2024. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfrmowOLD 3421* | Obsolete version of nfrmow 3407 as of 21-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 | ||
Theorem | cbvreu 3422* | Change the bound variable of a restricted unique existential quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker cbvreuw 3404 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrmo 3423* | Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker cbvrmow 3403, cbvrmovw 3397 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvrmov 3424* | Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | cbvreuv 3425* | Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3398 for a version without ax-13 2369, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker cbvreuvw 3398 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfrmod 3426 | Deduction version of nfrmo 3428. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfreud 3427 | Deduction version of nfreu 3429. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) | ||
Theorem | nfrmo 3428 | Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfrmow 3407 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 | ||
Theorem | nfreu 3429 | Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfreuw 3408 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 | ||
Syntax | crab 3430 | Extend class notation to include the restricted class abstraction (class builder). |
class {𝑥 ∈ 𝐴 ∣ 𝜑} | ||
Definition | df-rab 3431 |
Define a restricted class abstraction (class builder): {𝑥 ∈ 𝐴 ∣ 𝜑}
is the class of all sets 𝑥 in 𝐴 such that 𝜑(𝑥) is true.
Definition of [TakeutiZaring] p.
20.
For the interpretation given in the previous paragraph to be correct, we need to assume Ⅎ𝑥𝐴, which is the case as soon as 𝑥 and 𝐴 are disjoint, which is generally the case. If 𝐴 were to depend on 𝑥, then the interpretation would be less obvious (think of the two extreme cases 𝐴 = {𝑥} and 𝐴 = 𝑥, for instance). See also df-ral 3060. (Contributed by NM, 22-Nov-1994.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | ||
Theorem | rabbidva2 3432* | Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabbia2 3433 | Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒} | ||
Theorem | rabbiia 3434 | Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.) (Proof shortened by Wolf Lammen, 12-Jan-2025.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabbiiaOLD 3435 | Obsolete version of rabbiia 3434 as of 12-Jan-2025. (Contributed by NM, 22-May-1999.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabbii 3436 | Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3438. (Contributed by Peter Mazsa, 1-Nov-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabbidva 3437* | Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.) (Proof shortened by SN, 3-Dec-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbidv 3438* | Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabswap 3439 | Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴} | ||
Theorem | cbvrabv 3440* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2152 and ax-13 2369. (Revised by Steven Nguyen, 4-Dec-2022.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | rabeqcda 3441* | When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3442. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) | ||
Theorem | rabeqc 3442* | A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.) |
⊢ (𝑥 ∈ 𝐴 → 𝜑) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 | ||
Theorem | rabeqi 3443 | Equality theorem for restricted class abstractions. Inference form of rabeqf 3464. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2135, ax-11 2152, ax-12 2169. (Revised by Gino Giotto, 3-Jun-2024.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} | ||
Theorem | rabeq 3444* | Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2135, ax-11 2152, ax-12 2169. (Revised by Gino Giotto, 20-Aug-2023.) |
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | ||
Theorem | rabeqdv 3445* | Equality of restricted class abstractions. Deduction form of rabeq 3444. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) | ||
Theorem | rabeqbidva 3446* | Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabeqbidv 3447* | Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabrabi 3448* | Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2135, ax-11 2152 and ax-12 2169. (Revised by Gino Giotto, 12-Oct-2024.) |
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) ⇒ ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} | ||
Theorem | nfrab1 3449 | The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.) |
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | ||
Theorem | rabid 3450 | An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) |
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | rabidim1 3451 | Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) | ||
Theorem | reqabi 3452 | Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} ⇒ ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) | ||
Theorem | rabrab 3453 | Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.) |
⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} | ||
Theorem | rabrabiOLD 3454* | Obsolete version of rabrabi 3448 as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2135 and ax-11 2152. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) ⇒ ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} | ||
Theorem | rabbida4 3455 | Version of rabbidva2 3432 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabbida 3456 | Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3437 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2135, ax-11 2152. (Revised by Wolf Lammen, 14-Mar-2025.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabbid 3457 | Version of rabbidv 3438 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabeqd 3458 | Deduction form of rabeq 3444. Note that contrary to rabeq 3444 it has no disjoint variable condition. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) | ||
Theorem | rabeqbida 3459 | Version of rabeqbidva 3446 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | rabbi 3460 | Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3436. (Contributed by NM, 25-Nov-2013.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabid2f 3461 | An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rabid2 3462* | An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2024.) |
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rabid2OLD 3463* | Obsolete version of rabid2 3462 as of 24-Nov-2024. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rabeqf 3464 | Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) | ||
Theorem | cbvrabw 3465* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3471 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Theorem | nfrabw 3466* | A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3470 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} | ||
Theorem | nfrabwOLD 3467* | Obsolete version of nfrabw 3466 as of 23-Nov2024. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} | ||
Theorem | rabbidaOLD 3468 | Obsolete version of rabbida 3456 as of 14-Mar-2025. (Contributed by BJ, 27-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | rabeqiOLD 3469 | Obsolete version of rabeqi 3443 as of 3-Jun-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2135 and ax-11 2152. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} | ||
Theorem | nfrab 3470 | A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfrabw 3466 when possible. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} | ||
Theorem | cbvrab 3471 | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker cbvrabw 3465 when possible. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} | ||
Syntax | cvv 3472 | Extend class notation to include the universal class symbol. |
class V | ||
Theorem | vjust 3473 | Justification theorem for df-v 3474. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} | ||
Definition | df-v 3474 |
Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21.
Also Definition 2.9 of [Quine] p. 19. The
class V can be described
as the "class of all sets"; vprc 5314
proves that V is not itself a set
in ZF. We will frequently use the expression 𝐴 ∈ V as a short way
to
say "𝐴 is a set", and isset 3485 proves that this expression has the
same meaning as ∃𝑥𝑥 = 𝐴.
In well-founded set theories without urelements, like ZF, the class V is equal to the von Neumann universe. However, the letter "V" does not stand for "von Neumann". The letter "V" was used earlier by Peano in 1889 for the universe of sets, where the letter V is derived from the Latin word "Verum", referring to the true truth constant 𝑇. Peano's notation V was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. The class constant V is the first class constant introduced in this database. As a constant, as opposed to a variable, it cannot be substituted with anything, and in particular it is not part of any disjoint variable condition. For a general discussion of the theory of classes, see mmset.html#class 3485. See dfv2 3475 for an alternate definition. (Contributed by NM, 26-May-1993.) |
⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | ||
Theorem | dfv2 3475 | Alternate definition of the universal class (see df-v 3474). (Contributed by BJ, 30-Nov-2019.) |
⊢ V = {𝑥 ∣ ⊤} | ||
Theorem | vex 3476 | All setvar variables are sets (see isset 3485). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2823 but it uses more axioms. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2169. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.) |
⊢ 𝑥 ∈ V | ||
Theorem | vexOLD 3477 | Obsolete version of vex 3476 as of 4-Sep-2024. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2169. (Revised by SN, 28-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑥 ∈ V | ||
Theorem | elv 3478 | If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3476), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
⊢ (𝑥 ∈ V → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | elvd 3479 | If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3476) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3478. (Contributed by Peter Mazsa, 23-Oct-2018.) |
⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | el2v 3480 | If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3476), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | eqv 3481* | The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2135, ax-11 2152, ax-13 2369. (Revised by BJ, 10-Aug-2022.) |
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | eqvf 3482 | The universe contains every set. (Contributed by BJ, 15-Jul-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | abv 3483 | The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 36089) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2808, ax-8 2106. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.) |
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | ||
Theorem | abvALT 3484 | Alternate proof of abv 3483, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | ||
Theorem | isset 3485* |
Two ways to express that "𝐴 is a set": A class 𝐴 is a
member
of the universal class V (see df-v 3474)
if and only if the class
𝐴 exists (i.e., there exists some set
𝑥
equal to class 𝐴).
Theorem 6.9 of [Quine] p. 43.
A class 𝐴 which is not a set is called a proper class. Conventions: We will often use the expression "𝐴 ∈ V " to mean "𝐴 is a set", for example in uniex 7733. To make some theorems more readily applicable, we will also use the more general expression 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean "𝐴 is a set", typically in an antecedent, or in a hypothesis for theorems in deduction form (see for instance uniexg 7732 compared with uniex 7733). That this is more general is seen either by substitution (when the variable 𝑉 has no other occurrences), or by elex 3491. (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | ||
Theorem | issetft 3486 | Closed theorem form of isset 3485 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3539. (Contributed by Wolf Lammen, 9-Apr-2025.) |
⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)) | ||
Theorem | issetf 3487 | A version of isset 3485 that does not require 𝑥 and 𝐴 to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | ||
Theorem | isseti 3488* | A way to say "𝐴 is a set" (inference form). (Contributed by NM, 24-Jun-1993.) Remove dependencies on axioms. (Revised by BJ, 13-Jul-2019.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 𝑥 = 𝐴 | ||
Theorem | issetri 3489* | A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑥 𝑥 = 𝐴 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | eqvisset 3490 | A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3485 and issetri 3489. (Contributed by BJ, 27-Apr-2019.) |
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | ||
Theorem | elex 3491 | If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | ||
Theorem | elexi 3492 | If a class is a member of another class, then it is a set. Inference associated with elex 3491. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | elexd 3493 | If a class is a member of another class, then it is a set. Deduction associated with elex 3491. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | elex2OLD 3494* | Obsolete version of elex2 2810 as of 30-Nov-2024. (Contributed by Alan Sare, 25-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) | ||
Theorem | elex22 3495* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | ||
Theorem | prcnel 3496 | A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.) |
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) | ||
Theorem | ralv 3497 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | rexv 3498 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | reuv 3499 | A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | ||
Theorem | rmov 3500 | An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |