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Mirrors > Home > ILE Home > Th. List > ecexr | GIF version |
Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecexr | ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimag 4672 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)) | |
2 | 1 | ibi 165 | . . . 4 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
3 | df-ec 6108 | . . . 4 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 2, 3 | eleq2s 2132 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
5 | df-rex 2312 | . . . 4 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴)) | |
6 | simpl 102 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵}) | |
7 | velsn 3392 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
8 | 6, 7 | sylib 127 | . . . . 5 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵) |
9 | 8 | eximi 1491 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵) |
10 | 5, 9 | sylbi 114 | . . 3 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵) |
11 | 4, 10 | syl 14 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵) |
12 | isset 2561 | . 2 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵) | |
13 | 11, 12 | sylibr 137 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 Vcvv 2557 {csn 3375 class class class wbr 3764 “ cima 4348 [cec 6104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-ec 6108 |
This theorem is referenced by: relelec 6146 ecdmn0m 6148 |
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