Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neik0pk1imk0 Structured version   Visualization version   GIF version

Theorem neik0pk1imk0 43100
Description: Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)
Hypotheses
Ref Expression
neik0pk1imk0.bex (πœ‘ β†’ 𝐡 ∈ 𝑉)
neik0pk1imk0.n (πœ‘ β†’ 𝑁 ∈ (𝒫 𝒫 𝐡 ↑m 𝐡))
neik0pk1imk0.k0p (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (π‘β€˜π‘₯) β‰  βˆ…)
neik0pk1imk0.k1 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 π΅βˆ€π‘‘ ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)))
Assertion
Ref Expression
neik0pk1imk0 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 𝐡 ∈ (π‘β€˜π‘₯))
Distinct variable groups:   𝐡,𝑠,𝑑   𝑁,𝑠,𝑑   πœ‘,𝑠,π‘₯   π‘₯,𝑑
Allowed substitution hints:   πœ‘(𝑑)   𝐡(π‘₯)   𝑁(π‘₯)   𝑉(π‘₯,𝑑,𝑠)

Proof of Theorem neik0pk1imk0
StepHypRef Expression
1 neik0pk1imk0.k1 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 π΅βˆ€π‘‘ ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)))
2 neik0pk1imk0.bex . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ 𝑉)
3 pwidg 4621 . . . . . . 7 (𝐡 ∈ 𝑉 β†’ 𝐡 ∈ 𝒫 𝐡)
4 sseq2 4007 . . . . . . . . . 10 (𝑑 = 𝐡 β†’ (𝑠 βŠ† 𝑑 ↔ 𝑠 βŠ† 𝐡))
54anbi2d 627 . . . . . . . . 9 (𝑑 = 𝐡 β†’ ((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) ↔ (𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡)))
6 eleq1 2819 . . . . . . . . 9 (𝑑 = 𝐡 β†’ (𝑑 ∈ (π‘β€˜π‘₯) ↔ 𝐡 ∈ (π‘β€˜π‘₯)))
75, 6imbi12d 343 . . . . . . . 8 (𝑑 = 𝐡 β†’ (((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)) ↔ ((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
87rspcv 3607 . . . . . . 7 (𝐡 ∈ 𝒫 𝐡 β†’ (βˆ€π‘‘ ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)) β†’ ((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
92, 3, 83syl 18 . . . . . 6 (πœ‘ β†’ (βˆ€π‘‘ ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)) β†’ ((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
109ralimdv 3167 . . . . 5 (πœ‘ β†’ (βˆ€π‘  ∈ 𝒫 π΅βˆ€π‘‘ ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)) β†’ βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
1110ralimdv 3167 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 π΅βˆ€π‘‘ ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) β†’ 𝑑 ∈ (π‘β€˜π‘₯)) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
121, 11mpd 15 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)))
13 r19.23v 3180 . . . . . 6 (βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)) ↔ (βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)))
1413biimpi 215 . . . . 5 (βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)) β†’ (βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)))
1514a1i 11 . . . 4 (πœ‘ β†’ (βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)) β†’ (βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
1615ralimdv 3167 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)) β†’ βˆ€π‘₯ ∈ 𝐡 (βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯))))
1712, 16mpd 15 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)))
18 neik0pk1imk0.k0p . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (π‘β€˜π‘₯) β‰  βˆ…)
19 neik0pk1imk0.n . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑁 ∈ (𝒫 𝒫 𝐡 ↑m 𝐡))
20 elmapi 8845 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝒫 𝒫 𝐡 ↑m 𝐡) β†’ 𝑁:π΅βŸΆπ’« 𝒫 𝐡)
2119, 20syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑁:π΅βŸΆπ’« 𝒫 𝐡)
2221ffvelcdmda 7085 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘β€˜π‘₯) ∈ 𝒫 𝒫 𝐡)
2322elpwid 4610 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘β€˜π‘₯) βŠ† 𝒫 𝐡)
2423sseld 3980 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (𝑠 ∈ (π‘β€˜π‘₯) β†’ 𝑠 ∈ 𝒫 𝐡))
2524ancrd 550 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (𝑠 ∈ (π‘β€˜π‘₯) β†’ (𝑠 ∈ 𝒫 𝐡 ∧ 𝑠 ∈ (π‘β€˜π‘₯))))
2625eximdv 1918 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (βˆƒπ‘  𝑠 ∈ (π‘β€˜π‘₯) β†’ βˆƒπ‘ (𝑠 ∈ 𝒫 𝐡 ∧ 𝑠 ∈ (π‘β€˜π‘₯))))
27 n0 4345 . . . . . . . 8 ((π‘β€˜π‘₯) β‰  βˆ… ↔ βˆƒπ‘  𝑠 ∈ (π‘β€˜π‘₯))
28 df-rex 3069 . . . . . . . 8 (βˆƒπ‘  ∈ 𝒫 𝐡𝑠 ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘ (𝑠 ∈ 𝒫 𝐡 ∧ 𝑠 ∈ (π‘β€˜π‘₯)))
2926, 27, 283imtr4g 295 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((π‘β€˜π‘₯) β‰  βˆ… β†’ βˆƒπ‘  ∈ 𝒫 𝐡𝑠 ∈ (π‘β€˜π‘₯)))
3029imp 405 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (π‘β€˜π‘₯) β‰  βˆ…) β†’ βˆƒπ‘  ∈ 𝒫 𝐡𝑠 ∈ (π‘β€˜π‘₯))
31 elpwi 4608 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝐡 β†’ 𝑠 βŠ† 𝐡)
3224, 31syl6 35 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (𝑠 ∈ (π‘β€˜π‘₯) β†’ 𝑠 βŠ† 𝐡))
3332ralrimivw 3148 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) β†’ 𝑠 βŠ† 𝐡))
3433adantr 479 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (π‘β€˜π‘₯) β‰  βˆ…) β†’ βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) β†’ 𝑠 βŠ† 𝐡))
3530, 34r19.29imd 3116 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (π‘β€˜π‘₯) β‰  βˆ…) β†’ βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡))
3635ex 411 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((π‘β€˜π‘₯) β‰  βˆ… β†’ βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡)))
3736ralimdva 3165 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘β€˜π‘₯) β‰  βˆ… β†’ βˆ€π‘₯ ∈ 𝐡 βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡)))
3818, 37mpd 15 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡))
39 ralim 3084 . 2 (βˆ€π‘₯ ∈ 𝐡 (βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ 𝐡 ∈ (π‘β€˜π‘₯)) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡) β†’ βˆ€π‘₯ ∈ 𝐡 𝐡 ∈ (π‘β€˜π‘₯)))
4017, 38, 39sylc 65 1 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 𝐡 ∈ (π‘β€˜π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator