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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 4793 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)) | |
2 | 1 | ibi 175 | . . . 4 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
3 | df-ec 6310 | . . . 4 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 2, 3 | eleq2s 2183 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
5 | df-rex 2366 | . . . 4 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴)) | |
6 | simpl 108 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵}) | |
7 | velsn 3469 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
8 | 6, 7 | sylib 121 | . . . . 5 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵) |
9 | 8 | eximi 1537 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵) |
10 | 5, 9 | sylbi 120 | . . 3 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵) |
11 | 4, 10 | syl 14 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵) |
12 | isset 2628 | . 2 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵) | |
13 | 11, 12 | sylibr 133 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∃wex 1427 ∈ wcel 1439 ∃wrex 2361 Vcvv 2622 {csn 3452 class class class wbr 3853 “ cima 4457 [cec 6306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-br 3854 df-opab 3908 df-xp 4460 df-cnv 4462 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-ec 6310 |
This theorem is referenced by: relelec 6348 ecdmn0m 6350 |
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