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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 6612 | . . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | |
2 | xpexg 4774 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
3 | 2 | anidms 397 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
4 | ssexg 4169 | . . 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 2164 Vcvv 2760 ⊆ wss 3154 × cxp 4658 Er wer 6586 |
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-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-xp 4666 df-rel 4667 df-cnv 4668 df-dm 4670 df-rn 4671 df-er 6589 |
This theorem is referenced by: erexb 6614 qliftlem 6669 qusaddvallemg 12919 qusaddflemg 12920 qusaddval 12921 qusaddf 12922 qusmulval 12923 qusmulf 12924 qusgrp2 13186 eqgen 13300 qusrng 13457 qusring2 13565 |
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