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Theorem unbndrank 9880
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 9833 . . . . . . . 8 (rank‘𝑦) ∈ On
2 ontri1 6420 . . . . . . . 8 (((rank‘𝑦) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
31, 2mpan 690 . . . . . . 7 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝑦)))
43ralbidv 3176 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦)))
5 ralnex 3070 . . . . . 6 (∀𝑦𝐴 ¬ 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
64, 5bitrdi 287 . . . . 5 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦)))
76rexbiia 3090 . . . 4 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
8 rexnal 3098 . . . 4 (∃𝑥 ∈ On ¬ ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
97, 8bitri 275 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 ↔ ¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
10 bndrank 9879 . . 3 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
119, 10sylbir 235 . 2 (¬ ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦) → 𝐴 ∈ V)
1211con1i 147 1 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  Oncon0 6386  cfv 6563  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by: (None)
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