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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-prexg | GIF version |
Description: Proof of prexg 3966 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-prexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3470 | . . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵}) | |
2 | 1 | eleq1d 2147 | . . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V)) |
3 | bj-zfpair2 10701 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ V | |
4 | 2, 3 | vtoclg 2658 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V) |
5 | preq1 3469 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
6 | 5 | eleq1d 2147 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V)) |
7 | 4, 6 | syl5ib 152 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V)) |
8 | 7 | vtocleg 2669 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V)) |
9 | 8 | imp 122 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 {cpr 3399 |
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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-pr 3964 ax-bdor 10607 ax-bdeq 10611 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: bj-snexg 10703 bj-unex 10710 |
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