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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | inundifss 3501 | The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.) |
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 | ||
Theorem | disjdif2 3502 | The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.) |
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) | ||
Theorem | difun2 3503 | Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.) |
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) | ||
Theorem | undifss 3504 | Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) | ||
Theorem | ssdifin0 3505 | A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) | ||
Theorem | ssdifeq0 3506 | A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) | ||
Theorem | ssundifim 3507 | A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ∖ 𝐵) ⊆ 𝐶) | ||
Theorem | difdifdirss 3508 | Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶)) | ||
Theorem | uneqdifeqim 3509 | Two ways that 𝐴 and 𝐵 can "partition" 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵)) | ||
Theorem | r19.2m 3510* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.) |
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | r19.2mOLD 3511* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3510 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | r19.3rm 3512* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28m 3513* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.3rmv 3514* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.) |
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.9rmv 3515* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.) |
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28mv 3516* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.45mv 3517* | Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.44mv 3518* | Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))) | ||
Theorem | r19.27m 3519* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) | ||
Theorem | r19.27mv 3520* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) | ||
Theorem | rzal 3521* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexn0 3522* | Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3523). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) | ||
Theorem | rexm 3523* | Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | ||
Theorem | ralidm 3524* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | ral0 3525 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
⊢ ∀𝑥 ∈ ∅ 𝜑 | ||
Theorem | rgenm 3526* | Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑) ⇒ ⊢ ∀𝑥 ∈ 𝐴 𝜑 | ||
Theorem | ralf0 3527* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
⊢ ¬ 𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) | ||
Theorem | ralm 3528 | Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | raaanlem 3529* | Special case of raaan 3530 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) | ||
Theorem | raaan 3530* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | raaanv 3531* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | sbss 3532* | Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
⊢ ([𝑦 / 𝑥]𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴) | ||
Theorem | sbcssg 3533 | Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | dcun 3534 | The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) |
⊢ (𝜑 → DECID 𝑘 ∈ 𝐴) & ⊢ (𝜑 → DECID 𝑘 ∈ 𝐵) ⇒ ⊢ (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵)) | ||
Syntax | cif 3535 | Extend class notation to include the conditional operator. See df-if 3536 for a description. (In older databases this was denoted "ded".) |
class if(𝜑, 𝐴, 𝐵) | ||
Definition | df-if 3536* |
Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if
𝜑 then 𝐴 else 𝐵".
See iftrue 3540 and iffalse 3543 for its
values. In mathematical literature, this operator is rarely defined
formally but is implicit in informal definitions such as "let
f(x)=0 if
x=0 and 1/x otherwise."
In the absence of excluded middle, this will tend to be useful where 𝜑 is decidable (in the sense of df-dc 835). (Contributed by NM, 15-May-1999.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | ||
Theorem | dfif6 3537* | An alternate definition of the conditional operator df-if 3536 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | ||
Theorem | ifeq1 3538 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) | ||
Theorem | ifeq2 3539 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) | ||
Theorem | iftrue 3540 | Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | ||
Theorem | iftruei 3541 | Inference associated with iftrue 3540. (Contributed by BJ, 7-Oct-2018.) |
⊢ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 | ||
Theorem | iftrued 3542 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴) | ||
Theorem | iffalse 3543 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | ||
Theorem | iffalsei 3544 | Inference associated with iffalse 3543. (Contributed by BJ, 7-Oct-2018.) |
⊢ ¬ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵 | ||
Theorem | iffalsed 3545 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵) | ||
Theorem | ifnefalse 3546 | When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 3543 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷) | ||
Theorem | ifsbdc 3547 | Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.) |
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) & ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) | ||
Theorem | dfif3 3548* | Alternate definition of the conditional operator df-if 3536. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 𝐶 = {𝑥 ∣ 𝜑} ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) | ||
Theorem | ifssun 3549 | A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.) |
⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) | ||
Theorem | ifidss 3550 | A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.) |
⊢ if(𝜑, 𝐴, 𝐴) ⊆ 𝐴 | ||
Theorem | ifeq12 3551 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷)) | ||
Theorem | ifeq1d 3552 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
Theorem | ifeq2d 3553 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | ||
Theorem | ifeq12d 3554 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷)) | ||
Theorem | ifbi 3555 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | ||
Theorem | ifbid 3556 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) | ||
Theorem | ifbieq1d 3557 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) | ||
Theorem | ifbieq2i 3558 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ 𝐴 = 𝐵 ⇒ ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) | ||
Theorem | ifbieq2d 3559 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) | ||
Theorem | ifbieq12i 3560 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) | ||
Theorem | ifbieq12d 3561 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) | ||
Theorem | nfifd 3562 | Deduction version of nfif 3563. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) | ||
Theorem | nfif 3563 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵) | ||
Theorem | ifcldadc 3564 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifeq1dadc 3565 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
Theorem | ifeq2dadc 3566 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | ||
Theorem | ifbothdadc 3567 | A formula 𝜃 containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜂 ∧ 𝜑) → 𝜓) & ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) & ⊢ (𝜂 → DECID 𝜑) ⇒ ⊢ (𝜂 → 𝜃) | ||
Theorem | ifbothdc 3568 | A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃) | ||
Theorem | ifiddc 3569 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | ||
Theorem | eqifdc 3570 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
⊢ (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)))) | ||
Theorem | ifcldcd 3571 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifnotdc 3572 | Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.) |
⊢ (DECID 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) | ||
Theorem | ifandc 3573 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (DECID 𝜑 → if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)) | ||
Theorem | ifordc 3574 | Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (DECID 𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵))) | ||
Theorem | ifmdc 3575 | If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.) |
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑) | ||
Syntax | cpw 3576 | Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.) |
class 𝒫 𝐴 | ||
Theorem | pwjust 3577* | Soundness justification theorem for df-pw 3578. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} | ||
Definition | df-pw 3578* | Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if 𝐴 is { 3 , 5 , 7 }, then 𝒫 𝐴 is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | ||
Theorem | pweq 3579 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pweqi 3580 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝒫 𝐴 = 𝒫 𝐵 | ||
Theorem | pweqd 3581 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | elpw 3582 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | velpw 3583* | Setvar variable membership in a power class (common case). See elpw 3582. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | ||
Theorem | elpwg 3584 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwi 3585 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | elpwb 3586 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwid 3587 | An element of a power class is a subclass. Deduction form of elpwi 3585. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | elelpwi 3588 | If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶) | ||
Theorem | nfpw 3589 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥𝒫 𝐴 | ||
Theorem | pwidg 3590 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | ||
Theorem | pwid 3591 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ∈ 𝒫 𝐴 | ||
Theorem | pwss 3592* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Syntax | csn 3593 | Extend class notation to include singleton. |
class {𝐴} | ||
Syntax | cpr 3594 | Extend class notation to include unordered pair. |
class {𝐴, 𝐵} | ||
Syntax | ctp 3595 | Extend class notation to include unordered triplet. |
class {𝐴, 𝐵, 𝐶} | ||
Syntax | cop 3596 | Extend class notation to include ordered pair. |
class ⟨𝐴, 𝐵⟩ | ||
Syntax | cotp 3597 | Extend class notation to include ordered triple. |
class ⟨𝐴, 𝐵, 𝐶⟩ | ||
Theorem | snjust 3598* | Soundness justification theorem for df-sn 3599. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | ||
Definition | df-sn 3599* | Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3607. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | ||
Definition | df-pr 3600 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3669. For a more traditional definition, but requiring a dummy variable, see dfpr2 3612. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) |
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