Theorem rankc1 9844

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 9840	. . 3 ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)
3	2	biantru 530	. 2 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
4		iunss 5038	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
5	1	rankval4 9841	. . . 4 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
6	5	sseq1i 4003	. . 3 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
7		rankon 9769	. . . . 5 ⊢ (rank‘𝑥) ∈ On
8		rankon 9769	. . . . 5 ⊢ (rank‘∪ 𝐴) ∈ On
9	7, 8	onsucssi 7810	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
10	9	ralbii 3092	. . 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 3990	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3470   ⊆ wss 3941  ∪ cuni 4898  ∪ ciun 4987  suc csuc 6352  ‘cfv 6529  rankcrnk 9737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705  ax-reg 9566  ax-inf2 9615

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6286  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-ov 7393  df-om 7836  df-2nd 7955  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-rdg 8389  df-r1 9738  df-rank 9739

This theorem is referenced by: (None)