Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axun2 | GIF version |
Description: axun2 4190 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 10612 | . . . 4 ⊢ BOUNDED 𝑧 ∈ 𝑤 | |
2 | 1 | ax-bdex 10610 | . . 3 ⊢ BOUNDED ∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 |
3 | df-rex 2354 | . . . 4 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤)) | |
4 | exancom 1539 | . . . 4 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ 𝑧 ∈ 𝑤) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | |
5 | 3, 4 | bitri 182 | . . 3 ⊢ (∃𝑤 ∈ 𝑥 𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
6 | 2, 5 | bd0 10615 | . 2 ⊢ BOUNDED ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) |
7 | ax-un 4188 | . 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
8 | 6, 7 | bdbm1.3ii 10682 | 1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∀wal 1282 ∃wex 1421 ∃wrex 2349 |
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-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-un 4188 ax-bd0 10604 ax-bdex 10610 ax-bdel 10612 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-rex 2354 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |