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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-zfpair2 | GIF version |
Description: Proof of zfpair2 4222 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 14843 | . . . . 5 ⊢ BOUNDED 𝑤 = 𝑥 | |
2 | ax-bdeq 14843 | . . . . 5 ⊢ BOUNDED 𝑤 = 𝑦 | |
3 | 1, 2 | ax-bdor 14839 | . . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦) |
4 | ax-pr 4221 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
5 | 3, 4 | bdbm1.3ii 14914 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | dfcleq 2181 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
7 | vex 2752 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
8 | 7 | elpr 3625 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
9 | 8 | bibi2i 227 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 9 | albii 1480 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
11 | 6, 10 | bitri 184 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
12 | 11 | exbii 1615 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
13 | 5, 12 | mpbir 146 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
14 | 13 | issetri 2758 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∨ wo 709 ∀wal 1361 = wceq 1363 ∃wex 1502 ∈ wcel 2158 Vcvv 2749 {cpr 3605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-pr 4221 ax-bdor 14839 ax-bdeq 14843 ax-bdsep 14907 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-sn 3610 df-pr 3611 |
This theorem is referenced by: bj-prexg 14934 |
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