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Theorem r1pwALT 9847
Description: Alternate shorter proof of r1pw 9846 based on the additional axioms ax-reg 9593 and ax-inf2 9642. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
r1pwALT (𝐡 ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡)))

Proof of Theorem r1pwALT
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eleq1 2820 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ (𝑅1β€˜π΅) ↔ 𝐴 ∈ (𝑅1β€˜π΅)))
2 pweq 4616 . . . . . 6 (π‘₯ = 𝐴 β†’ 𝒫 π‘₯ = 𝒫 𝐴)
32eleq1d 2817 . . . . 5 (π‘₯ = 𝐴 β†’ (𝒫 π‘₯ ∈ (𝑅1β€˜suc 𝐡) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡)))
41, 3bibi12d 345 . . . 4 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ (𝑅1β€˜π΅) ↔ 𝒫 π‘₯ ∈ (𝑅1β€˜suc 𝐡)) ↔ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡))))
54imbi2d 340 . . 3 (π‘₯ = 𝐴 β†’ ((𝐡 ∈ On β†’ (π‘₯ ∈ (𝑅1β€˜π΅) ↔ 𝒫 π‘₯ ∈ (𝑅1β€˜suc 𝐡))) ↔ (𝐡 ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡)))))
6 vex 3477 . . . . . . 7 π‘₯ ∈ V
76rankr1a 9837 . . . . . 6 (𝐡 ∈ On β†’ (π‘₯ ∈ (𝑅1β€˜π΅) ↔ (rankβ€˜π‘₯) ∈ 𝐡))
8 eloni 6374 . . . . . . 7 (𝐡 ∈ On β†’ Ord 𝐡)
9 ordsucelsuc 7814 . . . . . . 7 (Ord 𝐡 β†’ ((rankβ€˜π‘₯) ∈ 𝐡 ↔ suc (rankβ€˜π‘₯) ∈ suc 𝐡))
108, 9syl 17 . . . . . 6 (𝐡 ∈ On β†’ ((rankβ€˜π‘₯) ∈ 𝐡 ↔ suc (rankβ€˜π‘₯) ∈ suc 𝐡))
117, 10bitrd 279 . . . . 5 (𝐡 ∈ On β†’ (π‘₯ ∈ (𝑅1β€˜π΅) ↔ suc (rankβ€˜π‘₯) ∈ suc 𝐡))
126rankpw 9844 . . . . . 6 (rankβ€˜π’« π‘₯) = suc (rankβ€˜π‘₯)
1312eleq1i 2823 . . . . 5 ((rankβ€˜π’« π‘₯) ∈ suc 𝐡 ↔ suc (rankβ€˜π‘₯) ∈ suc 𝐡)
1411, 13bitr4di 289 . . . 4 (𝐡 ∈ On β†’ (π‘₯ ∈ (𝑅1β€˜π΅) ↔ (rankβ€˜π’« π‘₯) ∈ suc 𝐡))
15 onsuc 7803 . . . . 5 (𝐡 ∈ On β†’ suc 𝐡 ∈ On)
166pwex 5378 . . . . . 6 𝒫 π‘₯ ∈ V
1716rankr1a 9837 . . . . 5 (suc 𝐡 ∈ On β†’ (𝒫 π‘₯ ∈ (𝑅1β€˜suc 𝐡) ↔ (rankβ€˜π’« π‘₯) ∈ suc 𝐡))
1815, 17syl 17 . . . 4 (𝐡 ∈ On β†’ (𝒫 π‘₯ ∈ (𝑅1β€˜suc 𝐡) ↔ (rankβ€˜π’« π‘₯) ∈ suc 𝐡))
1914, 18bitr4d 282 . . 3 (𝐡 ∈ On β†’ (π‘₯ ∈ (𝑅1β€˜π΅) ↔ 𝒫 π‘₯ ∈ (𝑅1β€˜suc 𝐡)))
205, 19vtoclg 3542 . 2 (𝐴 ∈ V β†’ (𝐡 ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡))))
21 elex 3492 . . . 4 (𝐴 ∈ (𝑅1β€˜π΅) β†’ 𝐴 ∈ V)
22 elex 3492 . . . . 5 (𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡) β†’ 𝒫 𝐴 ∈ V)
23 pwexb 7757 . . . . 5 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2422, 23sylibr 233 . . . 4 (𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡) β†’ 𝐴 ∈ V)
2521, 24pm5.21ni 377 . . 3 (Β¬ 𝐴 ∈ V β†’ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡)))
2625a1d 25 . 2 (Β¬ 𝐴 ∈ V β†’ (𝐡 ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡))))
2720, 26pm2.61i 182 1 (𝐡 ∈ On β†’ (𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝒫 𝐴 ∈ (𝑅1β€˜suc 𝐡)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  Vcvv 3473  π’« cpw 4602  Ord word 6363  Oncon0 6364  suc csuc 6366  β€˜cfv 6543  π‘…1cr1 9763  rankcrnk 9764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-reg 9593  ax-inf2 9642
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-r1 9765  df-rank 9766
This theorem is referenced by: (None)
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