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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 2824 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | ideqg 5505 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐴 ↔ 𝐴 = 𝐴)) | |
3 | 1, 2 | mpbiri 250 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 I cid 5248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-br 4873 df-opab 4935 df-id 5249 df-xp 5347 df-rel 5348 |
This theorem is referenced by: issetid 5508 opelidres 5644 fvi 6501 |
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