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Mirrors > Home > MPE Home > Th. List > erex | Structured version Visualization version GIF version |
Description: An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
erex | ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erssxp 8032 | . . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | |
2 | sqxpexg 7224 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
3 | ssexg 5029 | . . 3 ⊢ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V) | |
4 | 1, 2, 3 | syl2an 591 | . 2 ⊢ ((𝑅 Er 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ V) |
5 | 4 | ex 403 | 1 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 Vcvv 3414 ⊆ wss 3798 × cxp 5340 Er wer 8006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-dm 5352 df-rn 5353 df-er 8009 |
This theorem is referenced by: erexb 8034 qliftlem 8093 qshash 14933 qusaddvallem 16564 qusaddflem 16565 qusaddval 16566 qusaddf 16567 qusmulval 16568 qusmulf 16569 qusgrp2 17887 efgrelexlemb 18516 efgcpbllemb 18521 frgpuplem 18538 qusring2 18974 vitalilem2 23775 vitalilem3 23776 tgjustr 25786 |
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