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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssexd | GIF version |
Description: Bounded version of ssexd 4063. (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 13091 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
5 | 1, 2, 4 | syl2anc 408 | 1 ⊢ (𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 BOUNDED wbdc 13027 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-bdsep 13071 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-bdc 13028 |
This theorem is referenced by: (None) |
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