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Mirrors > Home > MPE Home > Th. List > erth2 | Structured version Visualization version GIF version |
Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
erth2.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
erth2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
Ref | Expression |
---|---|
erth2 | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erth2.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | 1 | ersymb 8028 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
3 | erth2.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
4 | 1, 3 | erth 8061 | . . 3 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐵]𝑅 = [𝐴]𝑅)) |
5 | eqcom 2832 | . . 3 ⊢ ([𝐵]𝑅 = [𝐴]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅) | |
6 | 4, 5 | syl6bb 279 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
7 | 2, 6 | bitrd 271 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 Er wer 8011 [cec 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-er 8014 df-ec 8016 |
This theorem is referenced by: qliftel 8100 |
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