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| Mirrors > Home > MPE Home > Th. List > ideq | Structured version Visualization version GIF version | ||
| Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.) | 
| Ref | Expression | 
|---|---|
| ideq.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| ideq | ⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ideq.1 | . 2 ⊢ 𝐵 ∈ V | |
| 2 | ideqg 5861 | . 2 ⊢ (𝐵 ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 Vcvv 3479 class class class wbr 5142 I cid 5576 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: dmi 5931 resieq 6007 iss 6052 elidinxp 6061 restidsing 6070 imai 6091 intasym 6134 asymref 6135 intirr 6137 poirr2 6143 cnvi 6160 xpdifid 6187 coi1 6281 dfpo2 6315 dffun2 6570 dffun2OLD 6571 dffv2 7003 isof1oidb 7345 idssen 9038 dflt2 13191 relexpindlem 15103 opsrtoslem2 22081 hausdiag 23654 hauseqlcld 23655 metustid 24568 ltgov 28606 ex-id 30454 dfso2 35756 idsset 35892 dfon3 35894 elfix 35905 dffix2 35907 sscoid 35915 dffun10 35916 elfuns 35917 brsingle 35919 brapply 35940 brsuccf 35943 dfrdg4 35953 bj-imdiridlem 37187 iss2 38346 undmrnresiss 43622 dffrege99 43980 ipo0 44473 ifr0 44474 fourierdlem42 46169 | 
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