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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 5836 | . . . 4 ⊢ Rel I | |
| 3 | 2 | brrelex1i 5741 | . . 3 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V) |
| 4 | 1, 3 | anim12ci 614 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
| 5 | eleq1 2829 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
| 6 | 5 | biimparc 479 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
| 7 | 6 | elexd 3504 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
| 8 | simpl 482 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉) | |
| 9 | 7, 8 | jca 511 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
| 10 | eqeq1 2741 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
| 11 | eqeq2 2749 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵)) | |
| 12 | df-id 5578 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
| 13 | 10, 11, 12 | brabg 5544 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
| 14 | 4, 9, 13 | pm5.21nd 802 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 I cid 5577 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: ideq 5863 ididg 5864 poleloe 6151 isof1oidb 7344 pltval 18377 tglngne 28558 tgelrnln 28638 opeldifid 32612 ideq2 38308 idinxpss 38313 inxpssidinxp 38317 idinxpssinxp 38318 cnvref5 38352 rnxrnidres 38402 cossid 38481 fourierdlem42 46164 |
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