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Theorem elhf2 35147
Description: Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
Hypothesis
Ref Expression
elhf2.1 𝐴 ∈ V
Assertion
Ref Expression
elhf2 (𝐴 ∈ Hf ↔ (rankβ€˜π΄) ∈ Ο‰)

Proof of Theorem elhf2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elhf 35146 . 2 (𝐴 ∈ Hf ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2 omon 7867 . . 3 (Ο‰ ∈ On ∨ Ο‰ = On)
3 nnon 7861 . . . . . . . . 9 (π‘₯ ∈ Ο‰ β†’ π‘₯ ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 9831 . . . . . . . . 9 (π‘₯ ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ π‘₯))
63, 5syl 17 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ π‘₯))
76adantl 483 . . . . . . 7 ((Ο‰ ∈ On ∧ π‘₯ ∈ Ο‰) β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ π‘₯))
8 elnn 7866 . . . . . . . . 9 (((rankβ€˜π΄) ∈ π‘₯ ∧ π‘₯ ∈ Ο‰) β†’ (rankβ€˜π΄) ∈ Ο‰)
98expcom 415 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ ((rankβ€˜π΄) ∈ π‘₯ β†’ (rankβ€˜π΄) ∈ Ο‰))
109adantl 483 . . . . . . 7 ((Ο‰ ∈ On ∧ π‘₯ ∈ Ο‰) β†’ ((rankβ€˜π΄) ∈ π‘₯ β†’ (rankβ€˜π΄) ∈ Ο‰))
117, 10sylbid 239 . . . . . 6 ((Ο‰ ∈ On ∧ π‘₯ ∈ Ο‰) β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) β†’ (rankβ€˜π΄) ∈ Ο‰))
1211rexlimdva 3156 . . . . 5 (Ο‰ ∈ On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) β†’ (rankβ€˜π΄) ∈ Ο‰))
13 peano2 7881 . . . . . . . 8 ((rankβ€˜π΄) ∈ Ο‰ β†’ suc (rankβ€˜π΄) ∈ Ο‰)
1413adantr 482 . . . . . . 7 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ suc (rankβ€˜π΄) ∈ Ο‰)
15 r1rankid 9854 . . . . . . . . . 10 (𝐴 ∈ V β†’ 𝐴 βŠ† (𝑅1β€˜(rankβ€˜π΄)))
164, 15mp1i 13 . . . . . . . . 9 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ 𝐴 βŠ† (𝑅1β€˜(rankβ€˜π΄)))
174elpw 4607 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝑅1β€˜(rankβ€˜π΄)) ↔ 𝐴 βŠ† (𝑅1β€˜(rankβ€˜π΄)))
1816, 17sylibr 233 . . . . . . . 8 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ 𝐴 ∈ 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
19 nnon 7861 . . . . . . . . . 10 ((rankβ€˜π΄) ∈ Ο‰ β†’ (rankβ€˜π΄) ∈ On)
20 r1suc 9765 . . . . . . . . . 10 ((rankβ€˜π΄) ∈ On β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
2119, 20syl 17 . . . . . . . . 9 ((rankβ€˜π΄) ∈ Ο‰ β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
2221adantr 482 . . . . . . . 8 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
2318, 22eleqtrrd 2837 . . . . . . 7 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄)))
24 fveq2 6892 . . . . . . . . 9 (π‘₯ = suc (rankβ€˜π΄) β†’ (𝑅1β€˜π‘₯) = (𝑅1β€˜suc (rankβ€˜π΄)))
2524eleq2d 2820 . . . . . . . 8 (π‘₯ = suc (rankβ€˜π΄) β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄))))
2625rspcev 3613 . . . . . . 7 ((suc (rankβ€˜π΄) ∈ Ο‰ ∧ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄))) β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2714, 23, 26syl2anc 585 . . . . . 6 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2827expcom 415 . . . . 5 (Ο‰ ∈ On β†’ ((rankβ€˜π΄) ∈ Ο‰ β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯)))
2912, 28impbid 211 . . . 4 (Ο‰ ∈ On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰))
304tz9.13 9786 . . . . . 6 βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯)
31 rankon 9790 . . . . . 6 (rankβ€˜π΄) ∈ On
3230, 312th 264 . . . . 5 (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ On)
33 rexeq 3322 . . . . . 6 (Ο‰ = On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯)))
34 eleq2 2823 . . . . . 6 (Ο‰ = On β†’ ((rankβ€˜π΄) ∈ Ο‰ ↔ (rankβ€˜π΄) ∈ On))
3533, 34bibi12d 346 . . . . 5 (Ο‰ = On β†’ ((βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰) ↔ (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ On)))
3632, 35mpbiri 258 . . . 4 (Ο‰ = On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰))
3729, 36jaoi 856 . . 3 ((Ο‰ ∈ On ∨ Ο‰ = On) β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰))
382, 37ax-mp 5 . 2 (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰)
391, 38bitri 275 1 (𝐴 ∈ Hf ↔ (rankβ€˜π΄) ∈ Ο‰)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  Oncon0 6365  suc csuc 6367  β€˜cfv 6544  Ο‰com 7855  π‘…1cr1 9757  rankcrnk 9758   Hf chf 35144
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  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-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-r1 9759  df-rank 9760  df-hf 35145
This theorem is referenced by:  elhf2g  35148  hfsn  35151
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