Theorem rankxplim 9906

Description: The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 9909 for the successor case. (Contributed by NM, 19-Sep-2006.)

Hypotheses

Ref	Expression
rankxplim.1	⊢ 𝐴 ∈ V
rankxplim.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
rankxplim	⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))

Proof of Theorem rankxplim

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwuni 4908	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 ∪ ⟨𝑥, 𝑦⟩
2		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	uniop 5491	. . . . . . . . . . 11 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	4	pweqi 4575	. . . . . . . . . 10 ⊢ 𝒫 ∪ ⟨𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
6	1, 5	sseqtri 3960	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7		pwuni 4908	. . . . . . . . . . 11 ⊢ {𝑥, 𝑦} ⊆ 𝒫 ∪ {𝑥, 𝑦}
8	2, 3	unipr 4886	. . . . . . . . . . . 12 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
9	8	pweqi 4575	. . . . . . . . . . 11 ⊢ 𝒫 ∪ {𝑥, 𝑦} = 𝒫 (𝑥 ∪ 𝑦)
10	7, 9	sseqtri 3960	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} ⊆ 𝒫 (𝑥 ∪ 𝑦)
11	10	sspwi 4571	. . . . . . . . 9 ⊢ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦)
12	6, 11	sstri 3921	. . . . . . . 8 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦)
13	2, 3	unex 7768	. . . . . . . . . . 11 ⊢ (𝑥 ∪ 𝑦) ∈ V
14	13	pwex 5345	. . . . . . . . . 10 ⊢ 𝒫 (𝑥 ∪ 𝑦) ∈ V
15	14	pwex 5345	. . . . . . . . 9 ⊢ 𝒫 𝒫 (𝑥 ∪ 𝑦) ∈ V
16	15	rankss 9873	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦) → (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)))
17	12, 16	ax-mp 5	. . . . . . 7 ⊢ (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦))
18		rankxplim.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
19	18	rankel 9861	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
20		rankxplim.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
21	20	rankel 9861	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
22	2, 3, 18, 20	rankelun 9899	. . . . . . . . . 10 ⊢ (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
23	19, 21, 22	syl2an 608	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
24	23	adantl 487	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
25		ranklim 9868	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
26		ranklim 9868	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
27	25, 26	bitrd 282	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
28	27	adantr 486	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
29	24, 28	mpbid 235	. . . . . . 7 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
30		rankon 9813	. . . . . . . 8 ⊢ (rank‘⟨𝑥, 𝑦⟩) ∈ On
31		rankon 9813	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
32		ontr2 6417	. . . . . . . 8 ⊢ (((rank‘⟨𝑥, 𝑦⟩) ∈ On ∧ (rank‘(𝐴 ∪ 𝐵)) ∈ On) → (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵))))
33	30, 31, 32	mp2an 705	. . . . . . 7 ⊢ (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)))
34	17, 29, 33	sylancr 599	. . . . . 6 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)))
35	30, 31	onsucssi 7859	. . . . . 6 ⊢ ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
36	34, 35	sylib 221	. . . . 5 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
37	36	ralrimivva 3208	. . . 4 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
38		fveq2 6893	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
39		suceq 6437	. . . . . . . 8 ⊢ ((rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩) → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
40	38, 39	syl 18	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
41	40	sseq1d 3943	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵))))
42	41	ralxp 5822	. . . . 5 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
43	18, 20	xpex 7774	. . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V
44	43	rankbnd 9895	. . . . 5 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
45	42, 44	bitr3i 280	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
46	37, 45	sylib 221	. . 3 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
47	46	adantr 486	. 2 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
48	18, 20	rankxpl 9902	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
49	48	adantl 487	. 2 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
50	47, 49	eqssd 3929	1 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079  Vcvv 3453   ⊆ wss 3899   ∪ cun 4151  ∅c0 4280  𝒫 cpw 4559  {cpr 4589  ⟨cop 4837  ∪ cuni 4869   × cxp 5652  Oncon0 6368  Lim wlim 6369  suc csuc 6370  ‘cfv 6546  rankcrnk 9777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-reg 9596  ax-inf2 9652

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-om 7885  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-r1 9778  df-rank 9779

This theorem is used by:  rankxplim3  9908