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Theorem elhf2 31959
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 31958 . 2 (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2 omon 7030 . . 3 (ω ∈ On ∨ ω = On)
3 nnon 7025 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 8651 . . . . . . . . 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 7029 . . . . . . . . 9 (((rank‘𝐴) ∈ 𝑥𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
98expcom 451 . . . . . . . 8 (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
109adantl 482 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ ω) → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
117, 10sylbid 230 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
1211rexlimdva 3025 . . . . 5 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
13 peano2 7040 . . . . . . . 8 ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
1413adantr 481 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15 r1rankid 8674 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
164, 15mp1i 13 . . . . . . . . 9 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
174elpw 4141 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1816, 17sylibr 224 . . . . . . . 8 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
19 nnon 7025 . . . . . . . . . 10 ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20 r1suc 8585 . . . . . . . . . 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 2701 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24 fveq2 6153 . . . . . . . . 9 (𝑥 = suc (rank‘𝐴) → (𝑅1𝑥) = (𝑅1‘suc (rank‘𝐴)))
2524eleq2d 2684 . . . . . . . 8 (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
2625rspcev 3298 . . . . . . 7 ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2714, 23, 26syl2anc 692 . . . . . 6 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2827expcom 451 . . . . 5 (ω ∈ On → ((rank‘𝐴) ∈ ω → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥)))
2912, 28impbid 202 . . . 4 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
304tz9.13 8606 . . . . . 6 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
31 rankon 8610 . . . . . 6 (rank‘𝐴) ∈ On
3230, 312th 254 . . . . 5 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)
33 rexeq 3131 . . . . . 6 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
34 eleq2 2687 . . . . . 6 (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
3533, 34bibi12d 335 . . . . 5 (ω = On → ((∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω) ↔ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)))
3632, 35mpbiri 248 . . . 4 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
3729, 36jaoi 394 . . 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 264 1 (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3189  wss 3559  𝒫 cpw 4135  Oncon0 5687  suc csuc 5689  cfv 5852  ωcom 7019  𝑅1cr1 8577  rankcrnk 8578   Hf chf 31956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-r1 8579  df-rank 8580  df-hf 31957
This theorem is referenced by:  elhf2g  31960  hfsn  31963
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