Theorem rankc2 9870

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
rankc2	⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankc2

Step	Hyp	Ref	Expression
1		pwuni 4950	. . . . 5 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		rankr1b.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
3	2	uniex 7735	. . . . . . 7 ⊢ ∪ 𝐴 ∈ V
4	3	pwex 5379	. . . . . 6 ⊢ 𝒫 ∪ 𝐴 ∈ V
5	4	rankss 9848	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴))
6	1, 5	ax-mp 5	. . . 4 ⊢ (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴)
7	3	rankpw 9842	. . . 4 ⊢ (rank‘𝒫 ∪ 𝐴) = suc (rank‘∪ 𝐴)
8	6, 7	sseqtri 4019	. . 3 ⊢ (rank‘𝐴) ⊆ suc (rank‘∪ 𝐴)
9	8	a1i 11	. 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) ⊆ suc (rank‘∪ 𝐴))
10	2	rankel 9838	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
11		eleq1 2819	. . . . 5 ⊢ ((rank‘𝑥) = (rank‘∪ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘∪ 𝐴) ∈ (rank‘𝐴)))
12	10, 11	syl5ibcom 244	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴)))
13	12	rexlimiv 3146	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴))
14		rankon 9794	. . . 4 ⊢ (rank‘∪ 𝐴) ∈ On
15		rankon 9794	. . . 4 ⊢ (rank‘𝐴) ∈ On
16	14, 15	onsucssi 7834	. . 3 ⊢ ((rank‘∪ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴))
17	13, 16	sylib 217	. 2 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → suc (rank‘∪ 𝐴) ⊆ (rank‘𝐴))
18	9, 17	eqssd 4000	1 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ∃wrex 3068  Vcvv 3472   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909  suc csuc 6367  ‘cfv 6544  rankcrnk 9762

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-reg 9591  ax-inf2 9640

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-r1 9763  df-rank 9764

This theorem is referenced by: (None)