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Mirrors > Home > ILE Home > Th. List > oprcl | GIF version |
Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
oprcl | ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex2 2626 | . 2 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → ∃𝑦 𝑦 ∈ 〈𝐴, 𝐵〉) | |
2 | df-op 3431 | . . . . . . 7 ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
3 | 2 | eleq2i 2149 | . . . . . 6 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}) |
4 | df-clab 2070 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) | |
5 | 3, 4 | bitri 182 | . . . . 5 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) |
6 | 3simpa 936 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
7 | 6 | sbimi 1689 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 5, 7 | sylbi 119 | . . . 4 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | nfv 1462 | . . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) | |
10 | 9 | sbf 1702 | . . . 4 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 8, 10 | sylib 120 | . . 3 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | 11 | exlimiv 1530 | . 2 ⊢ (∃𝑦 𝑦 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
13 | 1, 12 | syl 14 | 1 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 ∃wex 1422 ∈ wcel 1434 [wsb 1687 {cab 2069 Vcvv 2612 {csn 3422 {cpr 3423 〈cop 3425 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-v 2614 df-op 3431 |
This theorem is referenced by: opth1 4026 opth 4027 0nelop 4038 |
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