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Theorem elhf2 33533
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 33532 . 2 (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2 omon 7580 . . 3 (ω ∈ On ∨ ω = On)
3 nnon 7575 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 9253 . . . . . . . . 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 7579 . . . . . . . . 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 3281 . . . . 5 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
13 peano2 7591 . . . . . . . 8 ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
1413adantr 481 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15 r1rankid 9276 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
164, 15mp1i 13 . . . . . . . . 9 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
174elpw 4542 . . . . . . . . 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 7575 . . . . . . . . . 10 ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20 r1suc 9187 . . . . . . . . . 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 2913 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24 fveq2 6663 . . . . . . . . 9 (𝑥 = suc (rank‘𝐴) → (𝑅1𝑥) = (𝑅1‘suc (rank‘𝐴)))
2524eleq2d 2895 . . . . . . . 8 (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
2625rspcev 3620 . . . . . . 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 9208 . . . . . 6 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
31 rankon 9212 . . . . . 6 (rank‘𝐴) ∈ On
3230, 312th 265 . . . . 5 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)
33 rexeq 3404 . . . . . 6 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
34 eleq2 2898 . . . . . 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 851 . . 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 841   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  wss 3933  𝒫 cpw 4535  Oncon0 6184  suc csuc 6186  cfv 6348  ωcom 7569  𝑅1cr1 9179  rankcrnk 9180   Hf chf 33530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-r1 9181  df-rank 9182  df-hf 33531
This theorem is referenced by:  elhf2g  33534  hfsn  33537
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