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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssex | GIF version |
Description: Bounded version of ssex 4126. (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 3134 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | bdssex.bd | . . . 4 ⊢ BOUNDED 𝐴 | |
3 | bdssex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | bdinex2 13935 | . . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V |
5 | eleq1 2233 | . . 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 1348 ∈ wcel 2141 Vcvv 2730 ∩ cin 3120 ⊆ wss 3121 BOUNDED wbdc 13875 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-bdsep 13919 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-bdc 13876 |
This theorem is referenced by: bdssexi 13938 bdssexg 13939 bdfind 13981 |
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