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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 5851 | . 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 205 = wceq 1540 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 I cid 5573 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 |
This theorem is referenced by: dmi 5921 resieq 5992 iss 6035 elidinxp 6043 restidsing 6052 imai 6073 intasym 6116 asymref 6117 intirr 6119 poirr2 6125 cnvi 6141 xpdifid 6167 coi1 6261 dfpo2 6295 dffun2 6553 dffun2OLD 6554 dffv2 6986 isof1oidb 7324 idssen 8999 dflt2 13134 relexpindlem 15017 opsrtoslem2 21927 hausdiag 23468 hauseqlcld 23469 metustid 24382 ltgov 28280 ex-id 30119 dfso2 35194 idsset 35331 dfon3 35333 elfix 35344 dffix2 35346 sscoid 35354 dffun10 35355 elfuns 35356 brsingle 35358 brapply 35379 brsuccf 35382 dfrdg4 35392 bj-imdiridlem 36529 iss2 37676 undmrnresiss 42817 dffrege99 43175 ipo0 43670 ifr0 43671 fourierdlem42 45323 |
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