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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssex | GIF version |
Description: Bounded version of ssex 4073. (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 3089 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | bdssex.bd | . . . 4 ⊢ BOUNDED 𝐴 | |
3 | bdssex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | bdinex2 13269 | . . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V |
5 | eleq1 2203 | . . 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 1332 ∈ wcel 1481 Vcvv 2689 ∩ cin 3075 ⊆ wss 3076 BOUNDED wbdc 13209 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-bdc 13210 |
This theorem is referenced by: bdssexi 13272 bdssexg 13273 bdfind 13315 |
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