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Mirrors > Home > MPE Home > Th. List > ider | Structured version Visualization version GIF version |
Description: The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ider | ⊢ I Er V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | df-id 5573 | . 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | eqer 8734 | 1 ⊢ I Er V |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3474 I cid 5572 Er wer 8696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-er 8699 |
This theorem is referenced by: (None) |
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