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Mirrors > Home > MPE Home > Th. List > ideqg | Structured version Visualization version GIF version |
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ideqg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉) | |
2 | reli 5405 | . . . 4 ⊢ Rel I | |
3 | 2 | brrelexi 5315 | . . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | anim12ci 592 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
5 | eleq1 2827 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
6 | 5 | biimparc 505 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
7 | 6 | elexd 3354 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
8 | simpl 474 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉) | |
9 | 7, 8 | jca 555 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
10 | eqeq1 2764 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
11 | eqeq2 2771 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵)) | |
12 | df-id 5174 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
13 | 10, 11, 12 | brabg 5144 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
14 | 4, 9, 13 | pm5.21nd 979 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 class class class wbr 4804 I cid 5173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 |
This theorem is referenced by: ideq 5430 ididg 5431 restidsingOLD 5617 poleloe 5685 isof1oidb 6737 pltval 17161 tglngne 25644 tgelrnln 25724 opeldifid 29719 ideq2 34402 idinxpss 34407 inxpssidinxp 34410 idinxpssinxp 34411 rnxrnidres 34482 cossid 34553 fourierdlem42 40869 |
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