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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ideqg1 | Structured version Visualization version GIF version |
Description: For sets, the identity
relation is the same thing as equality.
(Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon,
27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ,
24-Dec-2023.)
TODO: delete once bj-ideqg 34452 is in the main section. |
Ref | Expression |
---|---|
bj-ideqg1 | ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq12 2835 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵)) | |
2 | df-id 5460 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | bj-brab2a1 34444 | . 2 ⊢ (𝐴 I 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) |
4 | simpr 487 | . . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
5 | elex 3512 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
6 | 5 | a1d 25 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ V)) |
7 | elex 3512 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
8 | eleq1a 2908 | . . . . . . 7 ⊢ (𝐵 ∈ V → (𝐴 = 𝐵 → 𝐴 ∈ V)) | |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝐴 = 𝐵 → 𝐴 ∈ V)) |
10 | 6, 9 | jaoi 853 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → 𝐴 ∈ V)) |
11 | eleq1 2900 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
12 | 5, 11 | syl5ibcom 247 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ V)) |
13 | 7 | a1d 25 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝐴 = 𝐵 → 𝐵 ∈ V)) |
14 | 12, 13 | jaoi 853 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → 𝐵 ∈ V)) |
15 | 10, 14 | jcad 515 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
16 | 15 | ancrd 554 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))) |
17 | 4, 16 | impbid2 228 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)) |
18 | 3, 17 | syl5bb 285 | 1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 I cid 5459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 |
This theorem is referenced by: (None) |
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