Theorem rankc1 9559

Description: A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)

Hypothesis

Ref	Expression
rankr1b.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
rankc1	⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankc1

Step	Hyp	Ref	Expression
1		rankr1b.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	rankuniss 9555	. . 3 ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)
3	2	biantru 529	. 2 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
4		iunss 4971	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
5	1	rankval4 9556	. . . 4 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
6	5	sseq1i 3945	. . 3 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
7		rankon 9484	. . . . 5 ⊢ (rank‘𝑥) ∈ On
8		rankon 9484	. . . . 5 ⊢ (rank‘∪ 𝐴) ∈ On
9	7, 8	onsucssi 7663	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
10	9	ralbii 3090	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
11	4, 6, 10	3bitr4ri 303	. 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘∪ 𝐴))
12		eqss 3932	. 2 ⊢ ((rank‘𝐴) = (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
13	3, 11, 12	3bitr4i 302	1 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  ∪ cuni 4836  ∪ ciun 4921  suc csuc 6253  ‘cfv 6418  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by: (None)