![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2671 | . 2 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → ∃𝑦 𝑦 ∈ 〈𝐴, 𝐵〉) | |
2 | df-op 3500 | . . . . . . 7 ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
3 | 2 | eleq2i 2179 | . . . . . 6 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}) |
4 | df-clab 2100 | . . . . . 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 959 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
7 | 6 | sbimi 1718 | . . . . 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 1489 | . . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) | |
10 | 9 | sbf 1731 | . . . 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 1558 | . 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 943 ∃wex 1449 ∈ wcel 1461 [wsb 1716 {cab 2099 Vcvv 2655 {csn 3491 {cpr 3492 〈cop 3494 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-v 2657 df-op 3500 |
This theorem is referenced by: opth1 4116 opth 4117 0nelop 4128 |
Copyright terms: Public domain | W3C validator |