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Theorem neik0pk1imk0 43101
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 4623 . . . . . . 7 (𝐡 ∈ 𝑉 β†’ 𝐡 ∈ 𝒫 𝐡)
4 sseq2 4009 . . . . . . . . . 10 (𝑑 = 𝐡 β†’ (𝑠 βŠ† 𝑑 ↔ 𝑠 βŠ† 𝐡))
54anbi2d 628 . . . . . . . . 9 (𝑑 = 𝐡 β†’ ((𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝑑) ↔ (𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡)))
6 eleq1 2820 . . . . . . . . 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 3609 . . . . . . 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 3168 . . . . 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 3168 . . . 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 3181 . . . . . 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 3168 . . 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 8846 . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘β€˜π‘₯) ∈ 𝒫 𝒫 𝐡)
2322elpwid 4612 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘β€˜π‘₯) βŠ† 𝒫 𝐡)
2423sseld 3982 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (𝑠 ∈ (π‘β€˜π‘₯) β†’ 𝑠 ∈ 𝒫 𝐡))
2524ancrd 551 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (𝑠 ∈ (π‘β€˜π‘₯) β†’ (𝑠 ∈ 𝒫 𝐡 ∧ 𝑠 ∈ (π‘β€˜π‘₯))))
2625eximdv 1919 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (βˆƒπ‘  𝑠 ∈ (π‘β€˜π‘₯) β†’ βˆƒπ‘ (𝑠 ∈ 𝒫 𝐡 ∧ 𝑠 ∈ (π‘β€˜π‘₯))))
27 n0 4347 . . . . . . . 8 ((π‘β€˜π‘₯) β‰  βˆ… ↔ βˆƒπ‘  𝑠 ∈ (π‘β€˜π‘₯))
28 df-rex 3070 . . . . . . . 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 406 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (π‘β€˜π‘₯) β‰  βˆ…) β†’ βˆƒπ‘  ∈ 𝒫 𝐡𝑠 ∈ (π‘β€˜π‘₯))
31 elpwi 4610 . . . . . . . . 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 3149 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) β†’ 𝑠 βŠ† 𝐡))
3433adantr 480 . . . . . 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 3117 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (π‘β€˜π‘₯) β‰  βˆ…) β†’ βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡))
3635ex 412 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((π‘β€˜π‘₯) β‰  βˆ… β†’ βˆƒπ‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ∧ 𝑠 βŠ† 𝐡)))
3736ralimdva 3166 . . 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 3085 . 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 395   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ↑m cmap 8823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-map 8825
This theorem is referenced by: (None)
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