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Mirrors > Home > ILE Home > Th. List > elpwg | GIF version |
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Ref | Expression |
---|---|
elpwg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2150 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)) | |
2 | sseq1 3047 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | vex 2622 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | elpw 3435 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
5 | 1, 2, 4 | vtoclbg 2680 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∈ wcel 1438 ⊆ wss 2999 𝒫 cpw 3429 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-in 3005 df-ss 3012 df-pw 3431 |
This theorem is referenced by: elpwi 3438 elpwb 3439 pwidg 3443 prsspwg 3596 elpw2g 3992 snelpwi 4039 prelpwi 4041 pwel 4045 eldifpw 4299 f1opw2 5850 2pwuninelg 6048 tfrlemibfn 6093 tfr1onlembfn 6109 tfrcllembfn 6122 elpmg 6421 fopwdom 6552 fiinopn 11601 |
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