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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 8738 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
3 | erth2.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
4 | 1, 3 | erth 8774 | . . 3 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐵]𝑅 = [𝐴]𝑅)) |
5 | eqcom 2735 | . . 3 ⊢ ([𝐵]𝑅 = [𝐴]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅) | |
6 | 4, 5 | bitrdi 287 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
7 | 2, 6 | bitrd 279 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 Er wer 8721 [cec 8722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-er 8724 df-ec 8726 |
This theorem is referenced by: qliftel 8818 qusker 33061 qsdrnglem2 33207 |
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