Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7640 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
4 | 2, 3 | erth 7678 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
5 | 1, 4 | mpbid 221 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 class class class wbr 4583 Er wer 7626 [cec 7627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-er 7629 df-ec 7631 |
This theorem is referenced by: erdisj 7681 qsel 7713 addsrmo 9773 mulsrmo 9774 qusgrp2 17356 frgpinv 18000 qustgpopn 21733 blpnfctr 22051 pi1inv 22660 pi1xfrf 22661 pi1xfr 22663 pi1xfrcnvlem 22664 pi1cof 22667 vitalilem3 23185 sconpi1 30475 |
Copyright terms: Public domain | W3C validator |