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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axun2 | GIF version |
Description: axun2 4420 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-axun2 | ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdel 13856 | . . . 4 ⊢ BOUNDED 𝑧 ∈ 𝑤 | |
2 | 1 | ax-bdex 13854 | . . 3 ⊢ BOUNDED ∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 |
3 | df-rex 2454 | . . . 4 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤)) | |
4 | exancom 1601 | . . . 4 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | |
5 | 3, 4 | bitri 183 | . . 3 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
6 | 2, 5 | bd0 13859 | . 2 ⊢ BOUNDED ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) |
7 | ax-un 4418 | . 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
8 | 6, 7 | bdbm1.3ii 13926 | 1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1346 ∃wex 1485 ∃wrex 2449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-14 2144 ax-un 4418 ax-bd0 13848 ax-bdex 13854 ax-bdel 13856 ax-bdsep 13919 |
This theorem depends on definitions: df-bi 116 df-rex 2454 |
This theorem is referenced by: (None) |
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