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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1 | GIF version |
Description: Bounded version of inex1 3891. (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 9966 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝐵 |
4 | 3 | bdzfauscl 10010 | . . . 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 2034 | . . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵))) | |
7 | elin 3126 | . . . . . . 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 1359 | . . . . 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 1496 | . . 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 2564 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ∩ cin 2916 BOUNDED wbdc 9960 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-bdc 9961 |
This theorem is referenced by: bdinex2 10020 bdinex1g 10021 bdpeano5 10068 |
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