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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 4992 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)) | |
2 | 1 | ibi 176 | . . . 4 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
3 | df-ec 6561 | . . . 4 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 2, 3 | eleq2s 2284 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴) |
5 | df-rex 2474 | . . . 4 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴)) | |
6 | simpl 109 | . . . . . 6 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵}) | |
7 | velsn 3624 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
8 | 6, 7 | sylib 122 | . . . . 5 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵) |
9 | 8 | eximi 1611 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵) |
10 | 5, 9 | sylbi 121 | . . 3 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵) |
11 | 4, 10 | syl 14 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵) |
12 | isset 2758 | . 2 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵) | |
13 | 11, 12 | sylibr 134 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∃wrex 2469 Vcvv 2752 {csn 3607 class class class wbr 4018 “ cima 4647 [cec 6557 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
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-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-ec 6561 |
This theorem is referenced by: relelec 6601 ecdmn0m 6603 |
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