![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3193 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | df-pw 3592 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
4 | 1, 2, 3 | elab2 2900 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 𝒫 cpw 3590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 |
This theorem is referenced by: velpw 3597 elpwg 3598 prsspw 3780 pwprss 3820 pwtpss 3821 pwv 3823 sspwuni 3986 iinpw 3992 iunpwss 3993 0elpw 4182 pwuni 4210 snelpw 4231 sspwb 4234 ssextss 4238 pwin 4300 pwunss 4301 iunpw 4498 xpsspw 4756 ssenen 6879 pw1ne3 7259 3nsssucpw1 7265 ioof 10001 tgdom 14029 distop 14042 epttop 14047 resttopon 14128 txuni2 14213 |
Copyright terms: Public domain | W3C validator |