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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-prexg | GIF version |
Description: Proof of prexg 4038 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 3520 | . . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵}) | |
2 | 1 | eleq1d 2156 | . . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V)) |
3 | bj-zfpair2 11801 | . . . . 5 ⊢ {𝑥, 𝑦} ∈ V | |
4 | 2, 3 | vtoclg 2679 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V) |
5 | preq1 3519 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
6 | 5 | eleq1d 2156 | . . . 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 2690 | . 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 1289 ∈ wcel 1438 Vcvv 2619 {cpr 3447 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-pr 4036 ax-bdor 11707 ax-bdeq 11711 ax-bdsep 11775 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 |
This theorem is referenced by: bj-snexg 11803 bj-unex 11810 |
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