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Theorem bndrank 9273
Description: Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
bndrank (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem bndrank
StepHypRef Expression
1 rankon 9227 . . . . . . . 8 (rank‘𝑦) ∈ On
21onordi 6298 . . . . . . 7 Ord (rank‘𝑦)
3 eloni 6204 . . . . . . 7 (𝑥 ∈ On → Ord 𝑥)
4 ordsucsssuc 7541 . . . . . . 7 ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
52, 3, 4sylancr 589 . . . . . 6 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
61onsuci 7556 . . . . . . 7 suc (rank‘𝑦) ∈ On
7 suceloni 7531 . . . . . . 7 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 r1ord3 9214 . . . . . . 7 ((suc (rank‘𝑦) ∈ On ∧ suc 𝑥 ∈ On) → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
96, 7, 8sylancr 589 . . . . . 6 (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
105, 9sylbid 242 . . . . 5 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11 vex 3500 . . . . . 6 𝑦 ∈ V
1211rankid 9265 . . . . 5 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13 ssel 3964 . . . . 5 ((𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥) → (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑥)))
1410, 12, 13syl6mpi 67 . . . 4 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥𝑦 ∈ (𝑅1‘suc 𝑥)))
1514ralimdv 3181 . . 3 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16 dfss3 3959 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17 fvex 6686 . . . . 5 (𝑅1‘suc 𝑥) ∈ V
1817ssex 5228 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
1916, 18sylbir 237 . . 3 (∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
2015, 19syl6 35 . 2 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V))
2120rexlimiv 3283 1 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wcel 2113  wral 3141  wrex 3142  Vcvv 3497  wss 3939  Ord word 6193  Oncon0 6194  suc csuc 6196  cfv 6358  𝑅1cr1 9194  rankcrnk 9195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-reg 9059  ax-inf2 9107
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-r1 9196  df-rank 9197
This theorem is referenced by:  unbndrank  9274  scottex  9317
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