Theorem bndrank 8877

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 8831	. . . . . . . 8 ⊢ (rank‘𝑦) ∈ On
2	1	onordi 5993	. . . . . . 7 ⊢ Ord (rank‘𝑦)
3		eloni 5894	. . . . . . 7 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordsucsssuc 7188	. . . . . . 7 ⊢ ((Ord (rank‘𝑦) ∧ Ord 𝑥) → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
5	2, 3, 4	sylancr 698	. . . . . 6 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 ↔ suc (rank‘𝑦) ⊆ suc 𝑥))
6	1	onsuci 7203	. . . . . . 7 ⊢ suc (rank‘𝑦) ∈ On
7		suceloni 7178	. . . . . . 7 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
8		r1ord3 8818	. . . . . . 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 698	. . . . . 6 ⊢ (𝑥 ∈ On → (suc (rank‘𝑦) ⊆ suc 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
10	5, 9	sylbid 230	. . . . 5 ⊢ (𝑥 ∈ On → ((rank‘𝑦) ⊆ 𝑥 → (𝑅1‘suc (rank‘𝑦)) ⊆ (𝑅1‘suc 𝑥)))
11		vex 3343	. . . . . 6 ⊢ 𝑦 ∈ V
12	11	rankid 8869	. . . . 5 ⊢ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))
13		ssel 3738	. . . . 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 3101	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥)))
16		dfss3 3733	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘suc 𝑥))
17		fvex 6362	. . . . 5 ⊢ (𝑅1‘suc 𝑥) ∈ V
18	17	ssex 4954	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘suc 𝑥) → 𝐴 ∈ V)
19	16, 18	sylbir 225	. . 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 3165	1 ⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  Vcvv 3340   ⊆ wss 3715  Ord word 5883  Oncon0 5884  suc csuc 5886  ‘cfv 6049  𝑅₁cr1 8798  rankcrnk 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-rank 8801

This theorem is referenced by:  unbndrank  8878  scottex  8921