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Mirrors > Home > NFE 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 ∈ ℘B ↔ A ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw.1 | . 2 ⊢ A ∈ V | |
2 | sseq1 3292 | . 2 ⊢ (x = A → (x ⊆ B ↔ A ⊆ B)) | |
3 | df-pw 3724 | . 2 ⊢ ℘B = {x ∣ x ⊆ B} | |
4 | 1, 2, 3 | elab2 2988 | 1 ⊢ (A ∈ ℘B ↔ A ⊆ B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∈ wcel 1710 Vcvv 2859 ⊆ wss 3257 ℘cpw 3722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pw 3724 |
This theorem is referenced by: elpwg 3729 prsspw 3878 pwpr 3883 pwtp 3884 pwv 3886 sspwuni 4051 iinpw 4054 iunpwss 4055 snelpwg 4114 snelpwi 4116 unipw 4117 sspwb 4118 pwadjoin 4119 elpw1 4144 dfpw2 4327 eqpw1relk 4479 nnadjoinpw 4521 tfinnnlem1 4533 pw1fnf1o 5855 mapexi 6003 mapval2 6018 mapsspm 6021 mapsspw 6022 enpw1pw 6075 enprmaplem5 6080 enprmaplem6 6081 |
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