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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssex | GIF version |
Description: Bounded version of ssex 4119. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdssex.bd | ⊢ BOUNDED 𝐴 |
bdssex.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
bdssex | ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3129 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | bdssex.bd | . . . 4 ⊢ BOUNDED 𝐴 | |
3 | bdssex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | bdinex2 13782 | . . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V |
5 | eleq1 2229 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V)) | |
6 | 4, 5 | mpbii 147 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V) |
7 | 1, 6 | sylbi 120 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1343 ∈ wcel 2136 Vcvv 2726 ∩ cin 3115 ⊆ wss 3116 BOUNDED wbdc 13722 |
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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 ax-bdsep 13766 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-in 3122 df-ss 3129 df-bdc 13723 |
This theorem is referenced by: bdssexi 13785 bdssexg 13786 bdfind 13828 |
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