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Theorem bndrank 9881
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 9835 . . . . . . . 8 (rank‘𝑦) ∈ On
21onordi 6495 . . . . . . 7 Ord (rank‘𝑦)
3 eloni 6394 . . . . . . 7 (𝑥 ∈ On → Ord 𝑥)
4 ordsucsssuc 7843 . . . . . . 7 ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
52, 3, 4sylancr 587 . . . . . 6 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
61onsuci 7859 . . . . . . 7 suc (rank‘𝑦) ∈ On
7 onsuc 7831 . . . . . . 7 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 r1ord3 9822 . . . . . . 7 ((suc (rank‘𝑦) ∈ On ∧ suc 𝑥 ∈ On) → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
96, 7, 8sylancr 587 . . . . . 6 (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
105, 9sylbid 240 . . . . 5 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11 vex 3484 . . . . . 6 𝑦 ∈ V
1211rankid 9873 . . . . 5 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13 ssel 3977 . . . . 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 3169 . . 3 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16 dfss3 3972 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17 fvex 6919 . . . . 5 (𝑅1‘suc 𝑥) ∈ V
1817ssex 5321 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
1916, 18sylbir 235 . . 3 (∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
2015, 19syl6 35 . 2 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V))
2120rexlimiv 3148 1 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  Ord word 6383  Oncon0 6384  suc csuc 6386  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  unbndrank  9882  scottex  9925
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