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Theorem r1pwALT 8872
Description: Alternate shorter proof of r1pw 8871 based on the additional axioms ax-reg 8652 and ax-inf2 8701. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
r1pwALT (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Proof of Theorem r1pwALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2837 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ (𝑅1𝐵) ↔ 𝐴 ∈ (𝑅1𝐵)))
2 pweq 4298 . . . . . 6 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
32eleq1d 2834 . . . . 5 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ (𝑅1‘suc 𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
41, 3bibi12d 334 . . . 4 (𝑥 = 𝐴 → ((𝑥 ∈ (𝑅1𝐵) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝐵)) ↔ (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
54imbi2d 329 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ On → (𝑥 ∈ (𝑅1𝐵) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝐵))) ↔ (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))))
6 vex 3352 . . . . . . 7 𝑥 ∈ V
76rankr1a 8862 . . . . . 6 (𝐵 ∈ On → (𝑥 ∈ (𝑅1𝐵) ↔ (rank‘𝑥) ∈ 𝐵))
8 eloni 5876 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
9 ordsucelsuc 7168 . . . . . . 7 (Ord 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ suc 𝐵))
108, 9syl 17 . . . . . 6 (𝐵 ∈ On → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ suc 𝐵))
117, 10bitrd 268 . . . . 5 (𝐵 ∈ On → (𝑥 ∈ (𝑅1𝐵) ↔ suc (rank‘𝑥) ∈ suc 𝐵))
126rankpw 8869 . . . . . 6 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
1312eleq1i 2840 . . . . 5 ((rank‘𝒫 𝑥) ∈ suc 𝐵 ↔ suc (rank‘𝑥) ∈ suc 𝐵)
1411, 13syl6bbr 278 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝑥) ∈ suc 𝐵))
15 suceloni 7159 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ On)
166pwex 4976 . . . . . 6 𝒫 𝑥 ∈ V
1716rankr1a 8862 . . . . 5 (suc 𝐵 ∈ On → (𝒫 𝑥 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝑥) ∈ suc 𝐵))
1815, 17syl 17 . . . 4 (𝐵 ∈ On → (𝒫 𝑥 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝑥) ∈ suc 𝐵))
1914, 18bitr4d 271 . . 3 (𝐵 ∈ On → (𝑥 ∈ (𝑅1𝐵) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝐵)))
205, 19vtoclg 3415 . 2 (𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
21 elex 3361 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 ∈ V)
22 elex 3361 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ V)
23 pwexb 7121 . . . . 5 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2422, 23sylibr 224 . . . 4 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
2521, 24pm5.21ni 366 . . 3 𝐴 ∈ V → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
2625a1d 25 . 2 𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
2720, 26pm2.61i 176 1 (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1630  wcel 2144  Vcvv 3349  𝒫 cpw 4295  Ord word 5865  Oncon0 5866  suc csuc 5868  cfv 6031  𝑅1cr1 8788  rankcrnk 8789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-reg 8652  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-r1 8790  df-rank 8791
This theorem is referenced by: (None)
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