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Theorem elhf2 35142
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 ↔ (rankβ€˜π΄) ∈ Ο‰)

Proof of Theorem elhf2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elhf 35141 . 2 (𝐴 ∈ Hf ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2 omon 7866 . . 3 (Ο‰ ∈ On ∨ Ο‰ = On)
3 nnon 7860 . . . . . . . . 9 (π‘₯ ∈ Ο‰ β†’ π‘₯ ∈ On)
4 elhf2.1 . . . . . . . . . 10 𝐴 ∈ V
54rankr1a 9830 . . . . . . . . 9 (π‘₯ ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ π‘₯))
63, 5syl 17 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ π‘₯))
76adantl 482 . . . . . . 7 ((Ο‰ ∈ On ∧ π‘₯ ∈ Ο‰) β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ π‘₯))
8 elnn 7865 . . . . . . . . 9 (((rankβ€˜π΄) ∈ π‘₯ ∧ π‘₯ ∈ Ο‰) β†’ (rankβ€˜π΄) ∈ Ο‰)
98expcom 414 . . . . . . . 8 (π‘₯ ∈ Ο‰ β†’ ((rankβ€˜π΄) ∈ π‘₯ β†’ (rankβ€˜π΄) ∈ Ο‰))
109adantl 482 . . . . . . 7 ((Ο‰ ∈ On ∧ π‘₯ ∈ Ο‰) β†’ ((rankβ€˜π΄) ∈ π‘₯ β†’ (rankβ€˜π΄) ∈ Ο‰))
117, 10sylbid 239 . . . . . 6 ((Ο‰ ∈ On ∧ π‘₯ ∈ Ο‰) β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) β†’ (rankβ€˜π΄) ∈ Ο‰))
1211rexlimdva 3155 . . . . 5 (Ο‰ ∈ On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) β†’ (rankβ€˜π΄) ∈ Ο‰))
13 peano2 7880 . . . . . . . 8 ((rankβ€˜π΄) ∈ Ο‰ β†’ suc (rankβ€˜π΄) ∈ Ο‰)
1413adantr 481 . . . . . . 7 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ suc (rankβ€˜π΄) ∈ Ο‰)
15 r1rankid 9853 . . . . . . . . . 10 (𝐴 ∈ V β†’ 𝐴 βŠ† (𝑅1β€˜(rankβ€˜π΄)))
164, 15mp1i 13 . . . . . . . . 9 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ 𝐴 βŠ† (𝑅1β€˜(rankβ€˜π΄)))
174elpw 4606 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝑅1β€˜(rankβ€˜π΄)) ↔ 𝐴 βŠ† (𝑅1β€˜(rankβ€˜π΄)))
1816, 17sylibr 233 . . . . . . . 8 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ 𝐴 ∈ 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
19 nnon 7860 . . . . . . . . . 10 ((rankβ€˜π΄) ∈ Ο‰ β†’ (rankβ€˜π΄) ∈ On)
20 r1suc 9764 . . . . . . . . . 10 ((rankβ€˜π΄) ∈ On β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
2119, 20syl 17 . . . . . . . . 9 ((rankβ€˜π΄) ∈ Ο‰ β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
2221adantr 481 . . . . . . . 8 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = 𝒫 (𝑅1β€˜(rankβ€˜π΄)))
2318, 22eleqtrrd 2836 . . . . . . 7 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄)))
24 fveq2 6891 . . . . . . . . 9 (π‘₯ = suc (rankβ€˜π΄) β†’ (𝑅1β€˜π‘₯) = (𝑅1β€˜suc (rankβ€˜π΄)))
2524eleq2d 2819 . . . . . . . 8 (π‘₯ = suc (rankβ€˜π΄) β†’ (𝐴 ∈ (𝑅1β€˜π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄))))
2625rspcev 3612 . . . . . . 7 ((suc (rankβ€˜π΄) ∈ Ο‰ ∧ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄))) β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2714, 23, 26syl2anc 584 . . . . . 6 (((rankβ€˜π΄) ∈ Ο‰ ∧ Ο‰ ∈ On) β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
2827expcom 414 . . . . 5 (Ο‰ ∈ On β†’ ((rankβ€˜π΄) ∈ Ο‰ β†’ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯)))
2912, 28impbid 211 . . . 4 (Ο‰ ∈ On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰))
304tz9.13 9785 . . . . . 6 βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯)
31 rankon 9789 . . . . . 6 (rankβ€˜π΄) ∈ On
3230, 312th 263 . . . . 5 (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ On)
33 rexeq 3321 . . . . . 6 (Ο‰ = On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯)))
34 eleq2 2822 . . . . . 6 (Ο‰ = On β†’ ((rankβ€˜π΄) ∈ Ο‰ ↔ (rankβ€˜π΄) ∈ On))
3533, 34bibi12d 345 . . . . 5 (Ο‰ = On β†’ ((βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰) ↔ (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ On)))
3632, 35mpbiri 257 . . . 4 (Ο‰ = On β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰))
3729, 36jaoi 855 . . 3 ((Ο‰ ∈ On ∨ Ο‰ = On) β†’ (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰))
382, 37ax-mp 5 . 2 (βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯) ↔ (rankβ€˜π΄) ∈ Ο‰)
391, 38bitri 274 1 (𝐴 ∈ Hf ↔ (rankβ€˜π΄) ∈ Ο‰)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  π’« cpw 4602  Oncon0 6364  suc csuc 6366  β€˜cfv 6543  Ο‰com 7854  π‘…1cr1 9756  rankcrnk 9757   Hf chf 35139
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-r1 9758  df-rank 9759  df-hf 35140
This theorem is referenced by:  elhf2g  35143  hfsn  35146
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