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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 5865 | . 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 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 I cid 5582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 |
This theorem is referenced by: dmi 5935 resieq 6011 iss 6055 elidinxp 6064 restidsing 6073 imai 6094 intasym 6138 asymref 6139 intirr 6141 poirr2 6147 cnvi 6164 xpdifid 6190 coi1 6284 dfpo2 6318 dffun2 6573 dffun2OLD 6574 dffv2 7004 isof1oidb 7344 idssen 9036 dflt2 13187 relexpindlem 15099 opsrtoslem2 22098 hausdiag 23669 hauseqlcld 23670 metustid 24583 ltgov 28620 ex-id 30463 dfso2 35735 idsset 35872 dfon3 35874 elfix 35885 dffix2 35887 sscoid 35895 dffun10 35896 elfuns 35897 brsingle 35899 brapply 35920 brsuccf 35923 dfrdg4 35933 bj-imdiridlem 37168 iss2 38326 undmrnresiss 43594 dffrege99 43952 ipo0 44445 ifr0 44446 fourierdlem42 46105 |
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