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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1 | GIF version |
Description: Bounded version of inex1 4152. (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 15059 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝐵 |
4 | 3 | bdzfauscl 15103 | . . . 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 2183 | . . . . 5 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵))) | |
7 | elin 3333 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
8 | 7 | bibi2i 227 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
9 | 8 | albii 1481 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
10 | 6, 9 | bitri 184 | . . . 4 ⊢ (𝑥 = (𝐴 ∩ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
11 | 10 | exbii 1616 | . . 3 ⊢ (∃𝑥 𝑥 = (𝐴 ∩ 𝐵) ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
12 | 5, 11 | mpbir 146 | . 2 ⊢ ∃𝑥 𝑥 = (𝐴 ∩ 𝐵) |
13 | 12 | issetri 2761 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∀wal 1362 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 ∩ cin 3143 BOUNDED wbdc 15053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-bdsep 15097 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-bdc 15054 |
This theorem is referenced by: bdinex2 15113 bdinex1g 15114 bdpeano5 15156 |
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