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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 4880 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)) | |
2 | 1 | ibi 175 | . . . 4 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
3 | df-ec 6424 | . . . 4 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 2, 3 | eleq2s 2232 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
5 | df-rex 2420 | . . . 4 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴)) | |
6 | simpl 108 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵}) | |
7 | velsn 3539 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
8 | 6, 7 | sylib 121 | . . . . 5 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵) |
9 | 8 | eximi 1579 | . . . 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 2687 | . 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 1331 ∃wex 1468 ∈ wcel 1480 ∃wrex 2415 Vcvv 2681 {csn 3522 class class class wbr 3924 “ cima 4537 [cec 6420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-ec 6424 |
This theorem is referenced by: relelec 6462 ecdmn0m 6464 |
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