Theorem rankc1 9860

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 9856	. . 3 ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)
3	2	biantru 539	. 2 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ∧ (rank‘∪ 𝐴) ⊆ (rank‘𝐴)))
4		iunss 5002	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
5	1	rankval4 9857	. . . 4 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
6	5	sseq1i 3958	. . 3 ⊢ ((rank‘𝐴) ⊆ (rank‘∪ 𝐴) ↔ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
7		rankon 9777	. . . . 5 ⊢ (rank‘𝑥) ∈ On
8		rankon 9777	. . . . 5 ⊢ (rank‘∪ 𝐴) ∈ On
9	7, 8	onsucssi 7835	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
10	9	ralbii 3108	. . 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 3945	. 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 3076  Vcvv 3450   ⊆ wss 3898  ∪ cuni 4866  ∪ ciun 4950  suc csuc 6353  ‘cfv 6527  rankcrnk 9745

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-reg 9564  ax-inf2 9620

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9746  df-rank 9747

This theorem is used by: (None)