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Theorem unbndrank 9259
 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 9212 . . . . . . . 8 (rank‘𝑦) ∈ On
2 ontri1 6197 . . . . . . . 8 (((rank‘𝑦) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
31, 2mpan 689 . . . . . . 7 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
43ralbidv 3165 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦)))
5 ralnex 3202 . . . . . 6 (∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
64, 5syl6bb 290 . . . . 5 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦)))
76rexbiia 3212 . . . 4 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
8 rexnal 3204 . . . 4 (∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
97, 8bitri 278 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
10 bndrank 9258 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
119, 10sylbir 238 . 2 (¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) → 𝐴 ∈ V)
1211con1i 149 1 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ⊆ wss 3884  Oncon0 6163  ‘cfv 6328  rankcrnk 9180 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182 This theorem is referenced by: (None)
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