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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 7798 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
4 | 2, 3 | erth 7834 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅)) |
5 | 1, 4 | mpbid 222 | 1 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 class class class wbr 4685 Er wer 7784 [cec 7785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-er 7787 df-ec 7789 |
This theorem is referenced by: erdisj 7837 qsel 7869 addsrmo 9932 mulsrmo 9933 qusgrp2 17580 frgpinv 18223 qustgpopn 21970 blpnfctr 22288 pi1inv 22898 pi1xfrf 22899 pi1xfr 22901 pi1xfrcnvlem 22902 pi1cof 22905 vitalilem3 23424 sconnpi1 31347 |
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