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Mirrors > Home > ILE Home > Th. List > ercl2 | GIF version |
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersym.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ersym.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Ref | Expression |
---|---|
ercl2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersym.1 | . 2 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | ersym.2 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | 1, 2 | ersym 6575 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐴) |
4 | 1, 3 | ercl 6574 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 class class class wbr 4021 Er wer 6560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-br 4022 df-opab 4083 df-xp 4653 df-rel 4654 df-cnv 4655 df-dm 4657 df-er 6563 |
This theorem is referenced by: qliftfun 6647 nqnq0pi 7472 qusgrp2 13078 |
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