Theorem rankxplim2 9776

Description: If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)

Hypotheses

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

Assertion

Ref	Expression
rankxplim2	⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))

Proof of Theorem rankxplim2

Step	Hyp	Ref	Expression
1		0ellim 6371	. . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2		n0i 4291	. . . 4 ⊢ (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
3	1, 2	syl 17	. . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4		df-ne 2926	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5		rankxplim.1	. . . . . . 7 ⊢ 𝐴 ∈ V
6		rankxplim.2	. . . . . . 7 ⊢ 𝐵 ∈ V
7	5, 6	xpex 7689	. . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V
8	7	rankeq0 9757	. . . . 5 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
9	8	notbii 320	. . . 4 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
10	4, 9	bitr2i 276	. . 3 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
11	3, 10	sylib 218	. 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12		limuni2 6370	. . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵)))
13		limuni2 6370	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → Lim ∪ ∪ (rank‘(𝐴 × 𝐵)))
14	12, 13	syl 17	. . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ ∪ (rank‘(𝐴 × 𝐵)))
15		rankuni 9759	. . . . . 6 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
16		rankuni 9759	. . . . . . 7 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
17	16	unieqi 4870	. . . . . 6 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
18	15, 17	eqtr2i 2753	. . . . 5 ⊢ ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘∪ ∪ (𝐴 × 𝐵))
19		unixp 6230	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
20	19	fveq2d 6826	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
21	18, 20	eqtrid 2776	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
22		limeq 6319	. . . 4 ⊢ (∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)) → (Lim ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
23	21, 22	syl 17	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (Lim ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
24	14, 23	imbitrid 244	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵))))
25	11, 24	mpcom 38	1 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∪ cun 3901  ∅c0 4284  ∪ cuni 4858   × cxp 5617  Lim wlim 6308  ‘cfv 6482  rankcrnk 9659

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-reg 9484  ax-inf2 9537

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-r1 9660  df-rank 9661

This theorem is referenced by:  rankxpsuc  9778