Theorem rankc1 9855

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 9851	. . 3 ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)
3	2	biantru 539	. 2 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
4		iunss 5007	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
5	1	rankval4 9852	. . . 4 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
6	5	sseq1i 3962	. . 3 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
7		rankon 9780	. . . . 5 ⊢ (rank‘𝑥) ∈ On
8		rankon 9780	. . . . 5 ⊢ (rank‘∪ 𝐴) ∈ On
9	7, 8	onsucssi 7840	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
10	9	ralbii 3110	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
11	4, 6, 10	3bitr4ri 307	. 2 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘∪ 𝐴))
12		eqss 3949	. 2 ⊢ ((rank‘𝐴) = (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
13	3, 11, 12	3bitr4i 306	1 ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4870  ∪ ciun 4954  suc csuc 6363  ‘cfv 6537  rankcrnk 9748

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-reg 9567  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749  df-rank 9750

This theorem is used by: (None)