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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1 | GIF version |
Description: Bounded version of inex1 4032. (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 12971 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝐵 |
4 | 3 | bdzfauscl 13015 | . . . 4 ⊢ (𝐴 ∈ V → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
6 | dfcleq 2111 | . . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵))) | |
7 | elin 3229 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
8 | 7 | bibi2i 226 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
9 | 8 | albii 1431 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
10 | 6, 9 | bitri 183 | . . . 4 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
11 | 10 | exbii 1569 | . . 3 ⊢ (∃𝑥 𝑥 = (𝐴 ∩ 𝐵) ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
12 | 5, 11 | mpbir 145 | . 2 ⊢ ∃𝑥 𝑥 = (𝐴 ∩ 𝐵) |
13 | 12 | issetri 2669 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1314 = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 ∩ cin 3040 BOUNDED wbdc 12965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-bdsep 13009 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-bdc 12966 |
This theorem is referenced by: bdinex2 13025 bdinex1g 13026 bdpeano5 13068 |
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