Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bdssexg | GIF version |
Description: Bounded version of ssexg 4037. (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 3091 | . . . 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 2663 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | bdssex 13027 | . . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V) |
6 | 2, 5 | vtoclg 2720 | . 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 1316 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 BOUNDED wbdc 12965 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-bdsep 13009 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-bdc 12966 |
This theorem is referenced by: bdssexd 13030 bdrabexg 13031 bdunexb 13045 |
Copyright terms: Public domain | W3C validator |