Theorem bndrank 9801

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 9755	. . . . . . . 8 ⊢ (rank‘𝑦) ∈ On
2	1	onordi 6461	. . . . . . 7 ⊢ Ord (rank‘𝑦)
3		eloni 6358	. . . . . . 7 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordsucsssuc 7805	. . . . . . 7 ⊢ ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
5	2, 3, 4	sylancr 596	. . . . . 6 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
6	1	onsuci 7821	. . . . . . 7 ⊢ suc (rank‘𝑦) ∈ On
7		onsuc 7795	. . . . . . 7 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		r1ord3 9742	. . . . . . 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 596	. . . . . 6 ⊢ (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
10	5, 9	sylbid 242	. . . . 5 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11		vex 3460	. . . . . 6 ⊢ 𝑦 ∈ V
12	11	rankid 9793	. . . . 5 ⊢ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13		ssel 3932	. . . . 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 3178	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16		dfss3 3927	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17		fvex 6882	. . . . 5 ⊢ (𝑅1‘suc 𝑥) ∈ V
18	17	ssex 5279	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
19	16, 18	sylbir 237	. . 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 3158	1 ⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ⊆ wss 3906  Ord word 6347  Oncon0 6348  suc csuc 6350  ‘cfv 6523  𝑅₁cr1 9722  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-r1 9724  df-rank 9725

This theorem is referenced by:  unbndrank  9802  scottex  9845  onvf1odlem4  35453