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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 2570 | . 2 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → ∃𝑦 𝑦 ∈ 〈𝐴, 𝐵〉) | |
2 | df-op 3384 | . . . . . . 7 ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
3 | 2 | eleq2i 2104 | . . . . . 6 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}) |
4 | df-clab 2027 | . . . . . 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 173 | . . . . 5 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})) |
6 | 3simpa 901 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
7 | 6 | sbimi 1647 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 5, 7 | sylbi 114 | . . . 4 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) | |
10 | 9 | sbf 1660 | . . . 4 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 8, 10 | sylib 127 | . . 3 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | 11 | exlimiv 1489 | . 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 97 ∧ w3a 885 ∃wex 1381 ∈ wcel 1393 [wsb 1645 {cab 2026 Vcvv 2557 {csn 3375 {cpr 3376 〈cop 3378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 df-op 3384 |
This theorem is referenced by: opth1 3973 opth 3974 0nelop 3985 |
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