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Mirrors > Home > MPE Home > Th. List > ididg | Structured version Visualization version GIF version |
Description: A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ididg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | ideqg 5720 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐴 ↔ 𝐴 = 𝐴)) | |
3 | 1, 2 | mpbiri 261 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 I cid 5454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 |
This theorem is referenced by: issetid 5723 opelidres 5863 fvi 6787 |
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