Theorem rankc1 9830

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 9826	. . 3 ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)
3	2	biantru 529	. 2 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
4		iunss 5012	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
5	1	rankval4 9827	. . . 4 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
6	5	sseq1i 3978	. . 3 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
7		rankon 9755	. . . . 5 ⊢ (rank‘𝑥) ∈ On
8		rankon 9755	. . . . 5 ⊢ (rank‘∪ 𝐴) ∈ On
9	7, 8	onsucssi 7820	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
10	9	ralbii 3076	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
11	4, 6, 10	3bitr4ri 304	. 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘∪ 𝐴))
12		eqss 3965	. 2 ⊢ ((rank‘𝐴) = (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
13	3, 11, 12	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ∪ cuni 4874  ∪ ciun 4958  suc csuc 6337  ‘cfv 6514  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724  df-rank 9725

This theorem is referenced by: (None)