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Theorem bndrank 8564
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 8518 . . . . . . . 8 (rank‘𝑦) ∈ On
21onordi 5735 . . . . . . 7 Ord (rank‘𝑦)
3 eloni 5636 . . . . . . 7 (𝑥 ∈ On → Ord 𝑥)
4 ordsucsssuc 6892 . . . . . . 7 ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
52, 3, 4sylancr 693 . . . . . 6 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
61onsuci 6907 . . . . . . 7 suc (rank‘𝑦) ∈ On
7 suceloni 6882 . . . . . . 7 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 r1ord3 8505 . . . . . . 7 ((suc (rank‘𝑦) ∈ On ∧ suc 𝑥 ∈ On) → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
96, 7, 8sylancr 693 . . . . . 6 (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
105, 9sylbid 228 . . . . 5 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11 vex 3175 . . . . . 6 𝑦 ∈ V
1211rankid 8556 . . . . 5 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13 ssel 3561 . . . . 5 ((𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥) → (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑥)))
1410, 12, 13syl6mpi 64 . . . 4 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥𝑦 ∈ (𝑅1‘suc 𝑥)))
1514ralimdv 2945 . . 3 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16 dfss3 3557 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17 fvex 6098 . . . . 5 (𝑅1‘suc 𝑥) ∈ V
1817ssex 4725 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
1916, 18sylbir 223 . . 3 (∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
2015, 19syl6 34 . 2 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V))
2120rexlimiv 3008 1 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  wss 3539  Ord word 5625  Oncon0 5626  suc csuc 5628  cfv 5790  𝑅1cr1 8485  rankcrnk 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-reg 8357  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-r1 8487  df-rank 8488
This theorem is referenced by:  unbndrank  8565  scottex  8608
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