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Theorem elhf2 32609
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 32608 . 2 (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2 omon 7242 . . 3 (ω ∈ On ∨ ω = On)
3 nnon 7237 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 8874 . . . . . . . . 9 (𝑥 ∈ On → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
63, 5syl 17 . . . . . . . 8 (𝑥 ∈ ω → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
76adantl 473 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
8 elnn 7241 . . . . . . . . 9 (((rank‘𝐴) ∈ 𝑥𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
98expcom 450 . . . . . . . 8 (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
109adantl 473 . . . . . . 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 3169 . . . . 5 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
13 peano2 7252 . . . . . . . 8 ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
1413adantr 472 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15 r1rankid 8897 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
164, 15mp1i 13 . . . . . . . . 9 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
174elpw 4308 . . . . . . . . 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 7237 . . . . . . . . . 10 ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20 r1suc 8808 . . . . . . . . . 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 472 . . . . . . . 8 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
2318, 22eleqtrrd 2842 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
24 fveq2 6353 . . . . . . . . 9 (𝑥 = suc (rank‘𝐴) → (𝑅1𝑥) = (𝑅1‘suc (rank‘𝐴)))
2524eleq2d 2825 . . . . . . . 8 (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
2625rspcev 3449 . . . . . . 7 ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2714, 23, 26syl2anc 696 . . . . . 6 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2827expcom 450 . . . . 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 8829 . . . . . 6 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
31 rankon 8833 . . . . . 6 (rank‘𝐴) ∈ On
3230, 312th 254 . . . . 5 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)
33 rexeq 3278 . . . . . 6 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
34 eleq2 2828 . . . . . 6 (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
3533, 34bibi12d 334 . . . . 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 393 . . 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 382  wa 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  wss 3715  𝒫 cpw 4302  Oncon0 5884  suc csuc 5886  cfv 6049  ωcom 7231  𝑅1cr1 8800  rankcrnk 8801   Hf chf 32606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-reg 8664  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-r1 8802  df-rank 8803  df-hf 32607
This theorem is referenced by:  elhf2g  32610  hfsn  32613
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