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Mirrors > Home > ILE Home > Th. List > elpw | GIF version |
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
elpw.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elpw | ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sseq1 3125 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | df-pw 3517 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
4 | 1, 2, 3 | elab2 2836 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 𝒫 cpw 3515 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 |
This theorem is referenced by: velpw 3522 elpwg 3523 prsspw 3700 pwprss 3740 pwtpss 3741 pwv 3743 sspwuni 3905 iinpw 3911 iunpwss 3912 0elpw 4096 pwuni 4124 snelpw 4143 sspwb 4146 ssextss 4150 pwin 4212 pwunss 4213 iunpw 4409 xpsspw 4659 ssenen 6753 ioof 9784 tgdom 12280 distop 12293 epttop 12298 resttopon 12379 txuni2 12464 |
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