Theorem bndrank 9592

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 ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem bndrank

Step	Hyp	Ref	Expression
1		rankon 9546	. . . . . . . 8 ⊢ (rank‘𝑦) ∈ On
2	1	onordi 6369	. . . . . . 7 ⊢ Ord (rank‘𝑦)
3		eloni 6274	. . . . . . 7 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordsucsssuc 7659	. . . . . . 7 ⊢ ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
5	2, 3, 4	sylancr 587	. . . . . 6 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
6	1	onsuci 7674	. . . . . . 7 ⊢ suc (rank‘𝑦) ∈ On
7		suceloni 7649	. . . . . . 7 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		r1ord3 9533	. . . . . . 7 ⊢ ((suc (rank‘𝑦) ∈ On ∧ suc 𝑥 ∈ On) → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
9	6, 7, 8	sylancr 587	. . . . . 6 ⊢ (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
10	5, 9	sylbid 239	. . . . 5 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11		vex 3435	. . . . . 6 ⊢ 𝑦 ∈ V
12	11	rankid 9584	. . . . 5 ⊢ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13		ssel 3919	. . . . 5 ⊢ ((𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥) → (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑥)))
14	10, 12, 13	syl6mpi 67	. . . 4 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → 𝑦 ∈ (𝑅1‘suc 𝑥)))
15	14	ralimdv 3106	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16		dfss3 3914	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17		fvex 6782	. . . . 5 ⊢ (𝑅1‘suc 𝑥) ∈ V
18	17	ssex 5249	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
19	16, 18	sylbir 234	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
20	15, 19	syl6 35	. 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V))
21	20	rexlimiv 3211	1 ⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ⊆ wss 3892  Ord word 6263  Oncon0 6264  suc csuc 6266  ‘cfv 6431  𝑅₁cr1 9513  rankcrnk 9514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-reg 9321  ax-inf2 9369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-om 7702  df-2nd 7819  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-r1 9515  df-rank 9516

This theorem is referenced by:  unbndrank  9593  scottex  9636