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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 5195 | . 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 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 I cid 4948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 |
This theorem is referenced by: dmi 5261 resieq 5327 iss 5367 restidsing 5377 restidsingOLD 5378 imai 5397 issref 5428 intasym 5430 asymref 5431 intirr 5433 poirr2 5439 cnvi 5456 xpdifid 5481 coi1 5568 dffv2 6181 isof1oidb 6474 resiexg 6994 idssen 7886 dflt2 11857 relexpindlem 13651 opsrtoslem2 19306 hausdiag 21258 hauseqlcld 21259 metustid 22169 ltgov 25292 ex-id 26683 dfso2 30897 dfpo2 30898 idsset 31167 dfon3 31169 elfix 31180 dffix2 31182 sscoid 31190 dffun10 31191 elfuns 31192 brsingle 31194 brapply 31215 brsuccf 31218 dfrdg4 31228 undmrnresiss 36929 dffrege99 37276 ipo0 37674 ifr0 37675 fourierdlem42 39042 |
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