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Mirrors > Home > MPE Home > Th. List > erthi | Structured version Visualization version GIF version |
Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
erthi.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
erthi.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Ref | Expression |
---|---|
erthi | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erthi.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | erthi.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
3 | 2, 1 | ercl 8755 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
4 | 2, 3 | erth 8795 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
5 | 1, 4 | mpbid 232 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 class class class wbr 5148 Er wer 8741 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-er 8744 df-ec 8746 |
This theorem is referenced by: erdisj 8798 qsel 8835 addsrmo 11111 mulsrmo 11112 qusgrp2 19089 frgpinv 19797 qustgpopn 24144 blpnfctr 24462 pi1inv 25099 pi1xfrf 25100 pi1xfr 25102 pi1xfrcnvlem 25103 pi1cof 25106 vitalilem3 25659 rloccring 33257 fracfld 33290 qsdrngilem 33502 zringfrac 33562 sconnpi1 35224 qsalrel 42260 |
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