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Theorem unbndrank 8961
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 8914 . . . . . . . 8 (rank‘𝑦) ∈ On
2 ontri1 5983 . . . . . . . 8 (((rank‘𝑦) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
31, 2mpan 673 . . . . . . 7 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
43ralbidv 3185 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦)))
5 ralnex 3191 . . . . . 6 (∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
64, 5syl6bb 278 . . . . 5 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦)))
76rexbiia 3239 . . . 4 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
8 rexnal 3193 . . . 4 (∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
97, 8bitri 266 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
10 bndrank 8960 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
119, 10sylbir 226 . 2 (¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) → 𝐴 ∈ V)
1211con1i 146 1 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wcel 2157  wral 3107  wrex 3108  Vcvv 3402  wss 3780  Oncon0 5949  cfv 6110  rankcrnk 8882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-reg 8745  ax-inf2 8794
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-om 7305  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-r1 8883  df-rank 8884
This theorem is referenced by: (None)
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