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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssexd | GIF version |
Description: Bounded version of ssexd 4119. (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 13679 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
5 | 1, 2, 4 | syl2anc 409 | 1 ⊢ (𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 Vcvv 2724 ⊆ wss 3114 BOUNDED wbdc 13615 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-bdsep 13659 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2726 df-in 3120 df-ss 3127 df-bdc 13616 |
This theorem is referenced by: (None) |
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