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Mirrors > Home > ILE Home > Th. List > erex | 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 6572 | . . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | |
2 | xpexg 4752 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
3 | 2 | anidms 397 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
4 | ssexg 4154 | . . 3 ⊢ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V) | |
5 | 1, 3, 4 | syl2an 289 | . 2 ⊢ ((𝑅 Er 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ V) |
6 | 5 | ex 115 | 1 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 Vcvv 2749 ⊆ wss 3141 × cxp 4636 Er wer 6546 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-rel 4645 df-cnv 4646 df-dm 4648 df-rn 4649 df-er 6549 |
This theorem is referenced by: erexb 6574 qliftlem 6627 qusaddvallemg 12771 qusaddflemg 12772 qusaddval 12773 qusaddf 12774 qusmulval 12775 qusmulf 12776 qusgrp2 13016 eqgen 13127 qusrng 13267 qusring2 13371 |
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