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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-zfpair2 | GIF version |
Description: Proof of zfpair2 4070 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-zfpair2 | ⊢ {𝑥, 𝑦} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdeq 12599 | . . . . 5 ⊢ BOUNDED 𝑤 = 𝑥 | |
2 | ax-bdeq 12599 | . . . . 5 ⊢ BOUNDED 𝑤 = 𝑦 | |
3 | 1, 2 | ax-bdor 12595 | . . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦) |
4 | ax-pr 4069 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
5 | 3, 4 | bdbm1.3ii 12670 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | dfcleq 2094 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
7 | vex 2644 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
8 | 7 | elpr 3495 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
9 | 8 | bibi2i 226 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 9 | albii 1414 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
11 | 6, 10 | bitri 183 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
12 | 11 | exbii 1552 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
13 | 5, 12 | mpbir 145 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
14 | 13 | issetri 2650 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 670 ∀wal 1297 = wceq 1299 ∃wex 1436 ∈ wcel 1448 Vcvv 2641 {cpr 3475 |
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-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-pr 4069 ax-bdor 12595 ax-bdeq 12599 ax-bdsep 12663 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 |
This theorem is referenced by: bj-prexg 12690 |
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