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Mirrors > Home > MPE Home > Th. List > elpw2g | Structured version Visualization version GIF version |
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.) |
Ref | Expression |
---|---|
elpw2g | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4610 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssexg 5324 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
3 | elpwg 4606 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimparc 481 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵) |
5 | 2, 4 | syldan 592 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 𝐵) |
6 | 5 | expcom 415 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝒫 𝐵)) |
7 | 1, 6 | impbid2 225 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 𝒫 cpw 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 df-pw 4605 |
This theorem is referenced by: elpw2 5346 elpwi2OLD 5348 pwnss 5350 difelpw 5352 rabelpw 5353 pw2f1olem 9076 fineqvlem 9262 elfir 9410 r1sscl 9780 tskwe 9945 dfac8alem 10024 acni2 10041 fin1ai 10288 fin2i 10290 fin23lem7 10311 fin23lem11 10312 isfin2-2 10314 fin23lem39 10345 isf34lem1 10367 isf34lem2 10368 isf34lem4 10372 isf34lem5 10373 fin1a2lem12 10406 canthnumlem 10643 tsken 10749 tskss 10753 gruss 10791 ismre 17534 mreintcl 17539 mremre 17548 submre 17549 mrcval 17554 mrccl 17555 mrcun 17566 ismri 17575 acsfiel 17598 isacs1i 17601 catcoppccl 18067 catcoppcclOLD 18068 acsdrsel 18496 acsdrscl 18499 acsficl 18500 pmtrval 19319 pmtrrn 19325 istopg 22397 uniopn 22399 iscld 22531 ntrval 22540 clsval 22541 discld 22593 mretopd 22596 neival 22606 isnei 22607 lpval 22643 restdis 22682 ordtbaslem 22692 ordtuni 22694 cndis 22795 tgcmp 22905 hauscmplem 22910 comppfsc 23036 elkgen 23040 xkoopn 23093 elqtop 23201 kqffn 23229 isfbas 23333 filss 23357 snfbas 23370 elfg 23375 ufilss 23409 fixufil 23426 cfinufil 23432 ufinffr 23433 ufilen 23434 fin1aufil 23436 flimclslem 23488 hauspwpwf1 23491 supnfcls 23524 flimfnfcls 23532 ptcmplem1 23556 tsmsfbas 23632 blfvalps 23889 blfps 23912 blf 23913 bcthlem5 24845 minveclem3b 24945 sigaclcuni 33116 sigaclcu2 33118 pwsiga 33128 erdsze2lem2 34195 cvmsval 34257 cvmsss2 34265 neibastop2lem 35245 tailf 35260 pibt2 36298 fin2so 36475 sdclem1 36611 elrfirn 41433 elrfirn2 41434 istopclsd 41438 nacsfix 41450 dnnumch1 41786 inpw 47503 |
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