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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssexd | GIF version |
Description: Bounded version of ssexd 3979. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdssexd.1 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
bdssexd.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
bdssexd.bd | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdssexd | ⊢ (𝜑 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdssexd.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | bdssexd.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
3 | bdssexd.bd | . . 3 ⊢ BOUNDED 𝐴 | |
4 | 3 | bdssexg 11795 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
5 | 1, 2, 4 | syl2anc 403 | 1 ⊢ (𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 Vcvv 2619 ⊆ wss 2999 BOUNDED wbdc 11731 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-bdsep 11775 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-in 3005 df-ss 3012 df-bdc 11732 |
This theorem is referenced by: (None) |
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