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Mirrors > Home > ILE Home > Th. List > ideq | 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 4760 | . 2 ⊢ (𝐵 ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1348 ∈ wcel 2141 Vcvv 2730 class class class wbr 3987 I cid 4271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 |
This theorem is referenced by: dmi 4824 resieq 4899 resiexg 4934 iss 4935 imai 4965 issref 4991 intasym 4993 asymref 4994 intirr 4995 poirr2 5001 cnvi 5013 coi1 5124 idssen 6751 |
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