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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axun2 | GIF version |
Description: axun2 4138 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 9276 | . . . 4 ⊢ BOUNDED z ∈ w | |
2 | 1 | ax-bdex 9274 | . . 3 ⊢ BOUNDED ∃w ∈ x z ∈ w |
3 | df-rex 2306 | . . . 4 ⊢ (∃w ∈ x z ∈ w ↔ ∃w(w ∈ x ∧ z ∈ w)) | |
4 | exancom 1496 | . . . 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 9279 | . 2 ⊢ BOUNDED ∃w(z ∈ w ∧ w ∈ x) |
7 | ax-un 4136 | . 2 ⊢ ∃y∀z(∃w(z ∈ w ∧ w ∈ x) → z ∈ y) | |
8 | 6, 7 | bdbm1.3ii 9346 | 1 ⊢ ∃y∀z(z ∈ y ↔ ∃w(z ∈ w ∧ w ∈ x)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1240 ∃wex 1378 ∃wrex 2301 |
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 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-un 4136 ax-bd0 9268 ax-bdex 9274 ax-bdel 9276 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-rex 2306 |
This theorem is referenced by: (None) |
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