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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 2737 | . 2 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → ∃𝑦 𝑦 ∈ 〈𝐴, 𝐵〉) | |
2 | df-op 3579 | . . . . . . 7 ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
3 | 2 | eleq2i 2231 | . . . . . 6 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}) |
4 | df-clab 2151 | . . . . . 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 183 | . . . . 5 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) |
6 | 3simpa 983 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
7 | 6 | sbimi 1751 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 5, 7 | sylbi 120 | . . . 4 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | nfv 1515 | . . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) | |
10 | 9 | sbf 1764 | . . . 4 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 8, 10 | sylib 121 | . . 3 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | 11 | exlimiv 1585 | . 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 103 ∧ w3a 967 ∃wex 1479 [wsb 1749 ∈ wcel 2135 {cab 2150 Vcvv 2721 {csn 3570 {cpr 3571 〈cop 3573 |
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-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-v 2723 df-op 3579 |
This theorem is referenced by: opth1 4208 opth 4209 0nelop 4220 |
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