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Theorem elhf2 34813
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 34812 . 2 (𝐴 ∈ Hf ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2 omon 7818 . . 3 (Ο‰ ∈ On ∨ Ο‰ = On)
3 nnon 7812 . . . . . . . . 9 (π‘₯ ∈ Ο‰ β†’ π‘₯ ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 9780 . . . . . . . . 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 7817 . . . . . . . . 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 3149 . . . . 5 (Ο‰ ∈ On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) β†’ (rankβ€˜π΄) ∈ Ο‰))
13 peano2 7831 . . . . . . . 8 ((rankβ€˜π΄) ∈ Ο‰ β†’ suc (rankβ€˜π΄) ∈ Ο‰)
1413adantr 482 . . . . . . 7 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ suc (rankβ€˜π΄) ∈ Ο‰)
15 r1rankid 9803 . . . . . . . . . 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 4568 . . . . . . . . 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 7812 . . . . . . . . . 10 ((rankβ€˜π΄) ∈ Ο‰ β†’ (rankβ€˜π΄) ∈ On)
20 r1suc 9714 . . . . . . . . . 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 6846 . . . . . . . . 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 3583 . . . . . . 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 9735 . . . . . 6 βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯)
31 rankon 9739 . . . . . 6 (rankβ€˜π΄) ∈ On
3230, 312th 264 . . . . 5 (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ On)
33 rexeq 3309 . . . . . 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 3070  Vcvv 3447   βŠ† wss 3914  π’« cpw 4564  Oncon0 6321  suc csuc 6323  β€˜cfv 6500  Ο‰com 7806  π‘…1cr1 9706  rankcrnk 9707   Hf chf 34810
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-r1 9708  df-rank 9709  df-hf 34811
This theorem is referenced by:  elhf2g  34814  hfsn  34817
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