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Theorem unbndrank 8649
Description: The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
unbndrank 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem unbndrank
StepHypRef Expression
1 rankon 8602 . . . . . . . 8 (rank‘𝑦) ∈ On
2 ontri1 5716 . . . . . . . 8 (((rank‘𝑦) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
31, 2mpan 705 . . . . . . 7 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
43ralbidv 2980 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦)))
5 ralnex 2986 . . . . . 6 (∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
64, 5syl6bb 276 . . . . 5 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦)))
76rexbiia 3033 . . . 4 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
8 rexnal 2989 . . . 4 (∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
97, 8bitri 264 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
10 bndrank 8648 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
119, 10sylbir 225 . 2 (¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) → 𝐴 ∈ V)
1211con1i 144 1 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555  Oncon0 5682  cfv 5847  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572
This theorem is referenced by: (None)
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