Theorem rankxplim2 9881

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 6427	. . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2		n0i 4333	. . . 4 ⊢ (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
3	1, 2	syl 17	. . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4		df-ne 2940	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5		rankxplim.1	. . . . . . 7 ⊢ 𝐴 ∈ V
6		rankxplim.2	. . . . . . 7 ⊢ 𝐵 ∈ V
7	5, 6	xpex 7744	. . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V
8	7	rankeq0 9862	. . . . 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 217	. 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12		limuni2 6426	. . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵)))
13		limuni2 6426	. . . 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 9864	. . . . . 6 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
16		rankuni 9864	. . . . . . 7 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
17	16	unieqi 4921	. . . . . 6 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
18	15, 17	eqtr2i 2760	. . . . 5 ⊢ ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘∪ ∪ (𝐴 × 𝐵))
19		unixp 6281	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
20	19	fveq2d 6895	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
21	18, 20	eqtrid 2783	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
22		limeq 6376	. . . 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 243	. 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 205   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  Vcvv 3473   ∪ cun 3946  ∅c0 4322  ∪ cuni 4908   × cxp 5674  Lim wlim 6365  ‘cfv 6543  rankcrnk 9764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-reg 9593  ax-inf2 9642

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-r1 9765  df-rank 9766

This theorem is referenced by:  rankxpsuc  9883