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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssexd | GIF version |
Description: Bounded version of ssexd 4158. (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 15114 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
5 | 1, 2, 4 | syl2anc 411 | 1 ⊢ (𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 BOUNDED wbdc 15050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-bdsep 15094 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-bdc 15051 |
This theorem is referenced by: (None) |
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