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Theorem elhf2 34061
 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 34060 . 2 (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2 omon 7596 . . 3 (ω ∈ On ∨ ω = On)
3 nnon 7591 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 9311 . . . . . . . . 9 (𝑥 ∈ On → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
63, 5syl 17 . . . . . . . 8 (𝑥 ∈ ω → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
76adantl 485 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ 𝑥))
8 elnn 7595 . . . . . . . . 9 (((rank‘𝐴) ∈ 𝑥𝑥 ∈ ω) → (rank‘𝐴) ∈ ω)
98expcom 417 . . . . . . . 8 (𝑥 ∈ ω → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
109adantl 485 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ ω) → ((rank‘𝐴) ∈ 𝑥 → (rank‘𝐴) ∈ ω))
117, 10sylbid 243 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ ω) → (𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
1211rexlimdva 3208 . . . . 5 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) → (rank‘𝐴) ∈ ω))
13 peano2 7607 . . . . . . . 8 ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω)
1413adantr 484 . . . . . . 7 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → suc (rank‘𝐴) ∈ ω)
15 r1rankid 9334 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
164, 15mp1i 13 . . . . . . . . 9 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
174elpw 4501 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1816, 17sylibr 237 . . . . . . . 8 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
19 nnon 7591 . . . . . . . . . 10 ((rank‘𝐴) ∈ ω → (rank‘𝐴) ∈ On)
20 r1suc 9245 . . . . . . . . . 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 484 . . . . . . . 8 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
2318, 22eleqtrrd 2855 . . . . . . 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 2837 . . . . . . . 8 (𝑥 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
2625rspcev 3543 . . . . . . 7 ((suc (rank‘𝐴) ∈ ω ∧ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2714, 23, 26syl2anc 587 . . . . . 6 (((rank‘𝐴) ∈ ω ∧ ω ∈ On) → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
2827expcom 417 . . . . 5 (ω ∈ On → ((rank‘𝐴) ∈ ω → ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥)))
2912, 28impbid 215 . . . 4 (ω ∈ On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
304tz9.13 9266 . . . . . 6 𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)
31 rankon 9270 . . . . . 6 (rank‘𝐴) ∈ On
3230, 312th 267 . . . . 5 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)
33 rexeq 3324 . . . . . 6 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥)))
34 eleq2 2840 . . . . . 6 (ω = On → ((rank‘𝐴) ∈ ω ↔ (rank‘𝐴) ∈ On))
3533, 34bibi12d 349 . . . . 5 (ω = On → ((∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω) ↔ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ On)))
3632, 35mpbiri 261 . . . 4 (ω = On → (∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥) ↔ (rank‘𝐴) ∈ ω))
3729, 36jaoi 854 . . 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 278 1 (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  𝒫 cpw 4497  Oncon0 6174  suc csuc 6176  ‘cfv 6340  ωcom 7585  𝑅1cr1 9237  rankcrnk 9238   Hf chf 34058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-reg 9102  ax-inf2 9150 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-r1 9239  df-rank 9240  df-hf 34059 This theorem is referenced by:  elhf2g  34062  hfsn  34065
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