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Theorem bndrank 9880
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 9834 . . . . . . . 8 (rank‘𝑦) ∈ On
21onordi 6486 . . . . . . 7 Ord (rank‘𝑦)
3 eloni 6385 . . . . . . 7 (𝑥 ∈ On → Ord 𝑥)
4 ordsucsssuc 7831 . . . . . . 7 ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
52, 3, 4sylancr 585 . . . . . 6 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
61onsuci 7847 . . . . . . 7 suc (rank‘𝑦) ∈ On
7 onsuc 7819 . . . . . . 7 (𝑥 ∈ On → suc 𝑥 ∈ On)
8 r1ord3 9821 . . . . . . 7 ((suc (rank‘𝑦) ∈ On ∧ suc 𝑥 ∈ On) → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
96, 7, 8sylancr 585 . . . . . 6 (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
105, 9sylbid 239 . . . . 5 (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11 vex 3465 . . . . . 6 𝑦 ∈ V
1211rankid 9872 . . . . 5 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13 ssel 3972 . . . . 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 3158 . . 3 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16 dfss3 3967 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17 fvex 6913 . . . . 5 (𝑅1‘suc 𝑥) ∈ V
1817ssex 5325 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
1916, 18sylbir 234 . . 3 (∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
2015, 19syl6 35 . 2 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V))
2120rexlimiv 3137 1 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  wss 3946  Ord word 6374  Oncon0 6375  suc csuc 6377  cfv 6553  𝑅1cr1 9801  rankcrnk 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-reg 9631  ax-inf2 9680
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-om 7876  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-r1 9803  df-rank 9804
This theorem is referenced by:  unbndrank  9881  scottex  9924
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