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Theorem elhf2 33520
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 ↔ (rank‘𝐴) ∈ ω)

Proof of Theorem elhf2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elhf 33519 . 2 (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2 omon 7582 . . 3 (ω ∈ On ∨ ω = On)
3 nnon 7577 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 9257 . . . . . . . . 9 (𝑥 ∈ On → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
63, 5syl 17 . . . . . . . 8 (𝑥 ∈ ω → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
76adantl 482 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
8 elnn 7581 . . . . . . . . 9 (((rank‘𝐴) ∈ 𝑥𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
98expcom 414 . . . . . . . 8 (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
109adantl 482 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ ω) → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
117, 10sylbid 241 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
1211rexlimdva 3288 . . . . 5 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
13 peano2 7593 . . . . . . . 8 ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
1413adantr 481 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15 r1rankid 9280 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
164, 15mp1i 13 . . . . . . . . 9 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
174elpw 4548 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1816, 17sylibr 235 . . . . . . . 8 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
19 nnon 7577 . . . . . . . . . 10 ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20 r1suc 9191 . . . . . . . . . 10 ((rank‘𝐴) ∈ On → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
2119, 20syl 17 . . . . . . . . 9 ((rank‘𝐴) ∈ ω → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
2221adantr 481 . . . . . . . 8 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
2318, 22eleqtrrd 2920 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24 fveq2 6666 . . . . . . . . 9 (𝑥 = suc (rank‘𝐴) → (𝑅1𝑥) = (𝑅1‘suc (rank‘𝐴)))
2524eleq2d 2902 . . . . . . . 8 (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
2625rspcev 3626 . . . . . . 7 ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2714, 23, 26syl2anc 584 . . . . . 6 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2827expcom 414 . . . . 5 (ω ∈ On → ((rank‘𝐴) ∈ ω → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥)))
2912, 28impbid 213 . . . 4 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
304tz9.13 9212 . . . . . 6 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
31 rankon 9216 . . . . . 6 (rank‘𝐴) ∈ On
3230, 312th 265 . . . . 5 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)
33 rexeq 3411 . . . . . 6 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
34 eleq2 2905 . . . . . 6 (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
3533, 34bibi12d 347 . . . . 5 (ω = On → ((∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω) ↔ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)))
3632, 35mpbiri 259 . . . 4 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
3729, 36jaoi 853 . . 3 ((ω ∈ On ∨ ω = On) → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
382, 37ax-mp 5 . 2 (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω)
391, 38bitri 276 1 (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  wrex 3143  Vcvv 3499  wss 3939  𝒫 cpw 4541  Oncon0 6188  suc csuc 6190  cfv 6351  ωcom 7571  𝑅1cr1 9183  rankcrnk 9184   Hf chf 33517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-reg 9048  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-r1 9185  df-rank 9186  df-hf 33518
This theorem is referenced by:  elhf2g  33521  hfsn  33524
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