![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6582 | . . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | |
2 | xpexg 4758 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
3 | 2 | anidms 397 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
4 | ssexg 4157 | . . 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 2160 Vcvv 2752 ⊆ wss 3144 × cxp 4642 Er wer 6556 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
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 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-dm 4654 df-rn 4655 df-er 6559 |
This theorem is referenced by: erexb 6584 qliftlem 6639 qusaddvallemg 12809 qusaddflemg 12810 qusaddval 12811 qusaddf 12812 qusmulval 12813 qusmulf 12814 qusgrp2 13055 eqgen 13166 qusrng 13312 qusring2 13416 |
Copyright terms: Public domain | W3C validator |