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Theorem bndrank 9879
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 9833 . . . . . . . 8 (rank‘𝑦) ∈ On
21onordi 6497 . . . . . . 7 Ord (rank‘𝑦)
3 eloni 6396 . . . . . . 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 9820 . . . . . . 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 3482 . . . . . 6 𝑦 ∈ V
1211rankid 9871 . . . . 5 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13 ssel 3989 . . . . 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 3167 . . 3 (𝑥 ∈ On → (∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16 dfss3 3984 . . . 4 (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17 fvex 6920 . . . . 5 (𝑅1‘suc 𝑥) ∈ V
1817ssex 5327 . . . 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 3146 1 (∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  Ord word 6385  Oncon0 6386  suc csuc 6388  cfv 6563  𝑅1cr1 9800  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:  unbndrank  9880  scottex  9923
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