Theorem rankc2 9629

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 4878	. . . . 5 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		rankr1b.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
3	2	uniex 7594	. . . . . . 7 ⊢ ∪ 𝐴 ∈ V
4	3	pwex 5303	. . . . . 6 ⊢ 𝒫 ∪ 𝐴 ∈ V
5	4	rankss 9607	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴))
6	1, 5	ax-mp 5	. . . 4 ⊢ (rank‘𝐴) ⊆ (rank‘𝒫 ∪ 𝐴)
7	3	rankpw 9601	. . . 4 ⊢ (rank‘𝒫 ∪ 𝐴) = suc (rank‘∪ 𝐴)
8	6, 7	sseqtri 3957	. . 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 9597	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
11		eleq1 2826	. . . . 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 3209	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘∪ 𝐴) ∈ (rank‘𝐴))
14		rankon 9553	. . . 4 ⊢ (rank‘∪ 𝐴) ∈ On
15		rankon 9553	. . . 4 ⊢ (rank‘𝐴) ∈ On
16	14, 15	onsucssi 7688	. . 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 3938	1 ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  suc csuc 6268  ‘cfv 6433  rankcrnk 9521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by: (None)