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 2615 | . 2 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → ∃𝑦 𝑦 ∈ 〈𝐴, 𝐵〉) | |
2 | df-op 3407 | . . . . . . 7 ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
3 | 2 | eleq2i 2145 | . . . . . 6 ⊢ (𝑦 ∈ 〈𝐴, 𝐵〉 ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}) |
4 | df-clab 2068 | . . . . . 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 935 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
7 | 6 | sbimi 1687 | . . . . 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 1461 | . . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) | |
10 | 9 | sbf 1700 | . . . 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 1529 | . 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 919 ∃wex 1421 ∈ wcel 1433 [wsb 1685 {cab 2067 Vcvv 2601 {csn 3398 {cpr 3399 〈cop 3401 |
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 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 df-op 3407 |
This theorem is referenced by: opth1 3991 opth 3992 0nelop 4003 |
Copyright terms: Public domain | W3C validator |