Theorem rankxplim 8742

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 8745 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 4474	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 ∪ ⟨𝑥, 𝑦⟩
2		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	uniop 4977	. . . . . . . . . . 11 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	4	pweqi 4162	. . . . . . . . . 10 ⊢ 𝒫 ∪ ⟨𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
6	1, 5	sseqtri 3637	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7		pwuni 4474	. . . . . . . . . . 11 ⊢ {𝑥, 𝑦} ⊆ 𝒫 ∪ {𝑥, 𝑦}
8	2, 3	unipr 4449	. . . . . . . . . . . 12 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
9	8	pweqi 4162	. . . . . . . . . . 11 ⊢ 𝒫 ∪ {𝑥, 𝑦} = 𝒫 (𝑥 ∪ 𝑦)
10	7, 9	sseqtri 3637	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} ⊆ 𝒫 (𝑥 ∪ 𝑦)
11		sspwb 4917	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝒫 (𝑥 ∪ 𝑦) ↔ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦))
12	10, 11	mpbi 220	. . . . . . . . 9 ⊢ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦)
13	6, 12	sstri 3612	. . . . . . . 8 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦)
14	2, 3	unex 6956	. . . . . . . . . . 11 ⊢ (𝑥 ∪ 𝑦) ∈ V
15	14	pwex 4848	. . . . . . . . . 10 ⊢ 𝒫 (𝑥 ∪ 𝑦) ∈ V
16	15	pwex 4848	. . . . . . . . 9 ⊢ 𝒫 𝒫 (𝑥 ∪ 𝑦) ∈ V
17	16	rankss 8712	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦) → (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)))
18	13, 17	ax-mp 5	. . . . . . 7 ⊢ (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦))
19		rankxplim.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
20	19	rankel 8702	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
21		rankxplim.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
22	21	rankel 8702	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
23	2, 3, 19, 21	rankelun 8735	. . . . . . . . . 10 ⊢ (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
24	20, 22, 23	syl2an 494	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
25	24	adantl 482	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
26		ranklim 8707	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
27		ranklim 8707	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
28	26, 27	bitrd 268	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
29	28	adantr 481	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
30	25, 29	mpbid 222	. . . . . . 7 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
31		rankon 8658	. . . . . . . 8 ⊢ (rank‘⟨𝑥, 𝑦⟩) ∈ On
32		rankon 8658	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
33		ontr2 5772	. . . . . . . 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))))
34	31, 32, 33	mp2an 708	. . . . . . 7 ⊢ (((rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∧ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)))
35	18, 30, 34	sylancr 695	. . . . . 6 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)))
36	31, 32	onsucssi 7041	. . . . . 6 ⊢ ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
37	35, 36	sylib 208	. . . . 5 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
38	37	ralrimivva 2971	. . . 4 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
39		fveq2 6191	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
40		suceq 5790	. . . . . . . 8 ⊢ ((rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩) → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
41	39, 40	syl 17	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
42	41	sseq1d 3632	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵))))
43	42	ralxp 5263	. . . . 5 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
44	19, 21	xpex 6962	. . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V
45	44	rankbnd 8731	. . . . 5 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
46	43, 45	bitr3i 266	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
47	38, 46	sylib 208	. . 3 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
48	47	adantr 481	. 2 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
49	19, 21	rankxpl 8738	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
50	49	adantl 482	. 2 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
51	48, 50	eqssd 3620	1 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {cpr 4179  ⟨cop 4183  ∪ cuni 4436   × cxp 5112  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ‘cfv 5888  rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  rankxplim3  8744