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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axun2 | GIF version |
Description: axun2 4172 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 9941 | . . . 4 ⊢ BOUNDED 𝑧 ∈ 𝑤 | |
2 | 1 | ax-bdex 9939 | . . 3 ⊢ BOUNDED ∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 |
3 | df-rex 2312 | . . . 4 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤)) | |
4 | exancom 1499 | . . . 4 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | |
5 | 3, 4 | bitri 173 | . . 3 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
6 | 2, 5 | bd0 9944 | . 2 ⊢ BOUNDED ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) |
7 | ax-un 4170 | . 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
8 | 6, 7 | bdbm1.3ii 10011 | 1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃wex 1381 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-un 4170 ax-bd0 9933 ax-bdex 9939 ax-bdel 9941 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-rex 2312 |
This theorem is referenced by: (None) |
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