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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssexg | GIF version |
Description: Bounded version of ssexg 4126. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdssexg.bd | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdssexg | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3171 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵)) | |
2 | 1 | imbi1d 230 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 → 𝐴 ∈ V) ↔ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V))) |
3 | bdssexg.bd | . . . 4 ⊢ BOUNDED 𝐴 | |
4 | vex 2733 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | bdssex 13902 | . . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V) |
6 | 2, 5 | vtoclg 2790 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)) |
7 | 6 | impcom 124 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 BOUNDED wbdc 13840 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-bdsep 13884 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-bdc 13841 |
This theorem is referenced by: bdssexd 13905 bdrabexg 13906 bdunexb 13920 |
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