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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 8303 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
4 | 2, 3 | erth 8341 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
5 | 1, 4 | mpbid 234 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 class class class wbr 5069 Er wer 8289 [cec 8290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-er 8292 df-ec 8294 |
This theorem is referenced by: erdisj 8344 qsel 8379 addsrmo 10498 mulsrmo 10499 qusgrp2 18220 frgpinv 18893 qustgpopn 22731 blpnfctr 23049 pi1inv 23659 pi1xfrf 23660 pi1xfr 23662 pi1xfrcnvlem 23663 pi1cof 23666 vitalilem3 24214 sconnpi1 32490 qsalrel 39131 |
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