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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1 | Structured version GIF version |
Description: Bounded version of inex1 3864. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdinex1.bd | ⊢ BOUNDED B |
bdinex1.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
bdinex1 | ⊢ (A ∩ B) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdinex1.1 | . . . 4 ⊢ A ∈ V | |
2 | bdinex1.bd | . . . . . 6 ⊢ BOUNDED B | |
3 | 2 | bdeli 6421 | . . . . 5 ⊢ BOUNDED y ∈ B |
4 | 3 | bdzfauscl 6460 | . . . 4 ⊢ (A ∈ V → ∃x∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B))) |
5 | 1, 4 | ax-mp 7 | . . 3 ⊢ ∃x∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B)) |
6 | dfcleq 2017 | . . . . 5 ⊢ (x = (A ∩ B) ↔ ∀y(y ∈ x ↔ y ∈ (A ∩ B))) | |
7 | elin 3102 | . . . . . . 7 ⊢ (y ∈ (A ∩ B) ↔ (y ∈ A ∧ y ∈ B)) | |
8 | 7 | bibi2i 216 | . . . . . 6 ⊢ ((y ∈ x ↔ y ∈ (A ∩ B)) ↔ (y ∈ x ↔ (y ∈ A ∧ y ∈ B))) |
9 | 8 | albii 1339 | . . . . 5 ⊢ (∀y(y ∈ x ↔ y ∈ (A ∩ B)) ↔ ∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B))) |
10 | 6, 9 | bitri 173 | . . . 4 ⊢ (x = (A ∩ B) ↔ ∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B))) |
11 | 10 | exbii 1480 | . . 3 ⊢ (∃x x = (A ∩ B) ↔ ∃x∀y(y ∈ x ↔ (y ∈ A ∧ y ∈ B))) |
12 | 5, 11 | mpbir 134 | . 2 ⊢ ∃x x = (A ∩ B) |
13 | 12 | issetri 2541 | 1 ⊢ (A ∩ B) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1226 = wceq 1228 ∃wex 1363 ∈ wcel 1375 Vcvv 2534 ∩ cin 2892 BOUNDED wbdc 6415 |
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-io 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-10 1378 ax-11 1379 ax-i12 1380 ax-bnd 1381 ax-4 1382 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-i5r 1411 ax-ext 2005 ax-bdsep 6455 |
This theorem depends on definitions: df-bi 110 df-tru 1231 df-nf 1330 df-sb 1629 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-v 2536 df-in 2900 df-bdc 6416 |
This theorem is referenced by: bdinex2 6467 bdinex1g 6468 bdpeano5 6510 |
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