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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 2180 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)) | |
2 | sseq1 3090 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | vex 2663 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | elpw 3486 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
5 | 1, 2, 4 | vtoclbg 2721 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1465 ⊆ wss 3041 𝒫 cpw 3480 |
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 |
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-pw 3482 |
This theorem is referenced by: elpwi 3489 elpwb 3490 pwidg 3494 prsspwg 3649 elpw2g 4051 snelpwi 4104 prelpwi 4106 pwel 4110 eldifpw 4368 f1opw2 5944 2pwuninelg 6148 tfrlemibfn 6193 tfr1onlembfn 6209 tfrcllembfn 6222 elpmg 6526 fopwdom 6698 fiinopn 12098 ssntr 12218 |
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