Theorem rankxplim 9891

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 9894 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 4921	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 ∪ ⟨𝑥, 𝑦⟩
2		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
3		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	2, 3	uniop 5490	. . . . . . . . . . 11 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	4	pweqi 4591	. . . . . . . . . 10 ⊢ 𝒫 ∪ ⟨𝑥, 𝑦⟩ = 𝒫 {𝑥, 𝑦}
6	1, 5	sseqtri 4007	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 {𝑥, 𝑦}
7		pwuni 4921	. . . . . . . . . . 11 ⊢ {𝑥, 𝑦} ⊆ 𝒫 ∪ {𝑥, 𝑦}
8	2, 3	unipr 4900	. . . . . . . . . . . 12 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
9	8	pweqi 4591	. . . . . . . . . . 11 ⊢ 𝒫 ∪ {𝑥, 𝑦} = 𝒫 (𝑥 ∪ 𝑦)
10	7, 9	sseqtri 4007	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} ⊆ 𝒫 (𝑥 ∪ 𝑦)
11	10	sspwi 4587	. . . . . . . . 9 ⊢ 𝒫 {𝑥, 𝑦} ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦)
12	6, 11	sstri 3968	. . . . . . . 8 ⊢ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 𝒫 (𝑥 ∪ 𝑦)
13	2, 3	unex 7736	. . . . . . . . . . 11 ⊢ (𝑥 ∪ 𝑦) ∈ V
14	13	pwex 5350	. . . . . . . . . 10 ⊢ 𝒫 (𝑥 ∪ 𝑦) ∈ V
15	14	pwex 5350	. . . . . . . . 9 ⊢ 𝒫 𝒫 (𝑥 ∪ 𝑦) ∈ V
16	15	rankss 9861	. . . . . . . 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 9851	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
20		rankxplim.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
21	20	rankel 9851	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (rank‘𝑦) ∈ (rank‘𝐵))
22	2, 3, 18, 20	rankelun 9884	. . . . . . . . . 10 ⊢ (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝑦) ∈ (rank‘𝐵)) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
23	19, 21, 22	syl2an 596	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
24	23	adantl 481	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
25		ranklim 9856	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
26		ranklim 9856	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
27	25, 26	bitrd 279	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ((rank‘(𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵))))
28	27	adantr 480	. . . . . . . 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 232	. . . . . . 7 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘𝒫 𝒫 (𝑥 ∪ 𝑦)) ∈ (rank‘(𝐴 ∪ 𝐵)))
30		rankon 9807	. . . . . . . 8 ⊢ (rank‘⟨𝑥, 𝑦⟩) ∈ On
31		rankon 9807	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
32		ontr2 6400	. . . . . . . 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 692	. . . . . . 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 587	. . . . . 6 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)))
35	30, 31	onsucssi 7834	. . . . . 6 ⊢ ((rank‘⟨𝑥, 𝑦⟩) ∈ (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
36	34, 35	sylib 218	. . . . 5 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
37	36	ralrimivva 3187	. . . 4 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)))
38		fveq2 6875	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩))
39		suceq 6419	. . . . . . . 8 ⊢ ((rank‘𝑧) = (rank‘⟨𝑥, 𝑦⟩) → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
40	38, 39	syl 17	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → suc (rank‘𝑧) = suc (rank‘⟨𝑥, 𝑦⟩))
41	40	sseq1d 3990	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵))))
42	41	ralxp 5821	. . . . 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 7745	. . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V
44	43	rankbnd 9880	. . . . 5 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)suc (rank‘𝑧) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
45	42, 44	bitr3i 277	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 suc (rank‘⟨𝑥, 𝑦⟩) ⊆ (rank‘(𝐴 ∪ 𝐵)) ↔ (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
46	37, 45	sylib 218	. . 3 ⊢ (Lim (rank‘(𝐴 ∪ 𝐵)) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
47	46	adantr 480	. 2 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) ⊆ (rank‘(𝐴 ∪ 𝐵)))
48	18, 20	rankxpl 9887	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
49	48	adantl 481	. 2 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
50	47, 49	eqssd 3976	1 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  {cpr 4603  ⟨cop 4607  ∪ cuni 4883   × cxp 5652  Oncon0 6352  Lim wlim 6353  suc csuc 6354  ‘cfv 6530  rankcrnk 9775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-reg 9604  ax-inf2 9653

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  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 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-r1 9776  df-rank 9777

This theorem is referenced by:  rankxplim3  9893