Theorem rankxplim2 9921

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 6442	. . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ∅ ∈ (rank‘(𝐴 × 𝐵)))
2		n0i 4289	. . . 4 ⊢ (∅ ∈ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
3	1, 2	syl 18	. . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
4		df-ne 2960	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
5		rankxplim.1	. . . . . . 7 ⊢ 𝐴 ∈ V
6		rankxplim.2	. . . . . . 7 ⊢ 𝐵 ∈ V
7	5, 6	xpex 7788	. . . . . 6 ⊢ (𝐴 × 𝐵) ∈ V
8	7	rankeq0 9901	. . . . 5 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
9	8	notbii 323	. . . 4 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
10	4, 9	bitr2i 279	. . 3 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
11	3, 10	sylib 221	. 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
12		limuni2 6441	. . . 4 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵)))
13		limuni2 6441	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → Lim ∪ ∪ (rank‘(𝐴 × 𝐵)))
14	12, 13	syl 18	. . 3 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ ∪ (rank‘(𝐴 × 𝐵)))
15		rankuni 9903	. . . . . 6 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
16		rankuni 9903	. . . . . . 7 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
17	16	unieqi 4882	. . . . . 6 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
18	15, 17	eqtr2i 2787	. . . . 5 ⊢ ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘∪ ∪ (𝐴 × 𝐵))
19		unixp 6300	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
20	19	fveq2d 6909	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
21	18, 20	eqtrid 2810	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
22		limeq 6389	. . . 4 ⊢ (∪ ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)) → (Lim ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
23	21, 22	syl 18	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (Lim ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
24	14, 23	imbitrid 247	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵))))
25	11, 24	mpcom 39	1 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ∪ cun 4152  ∅c0 4281  ∪ cuni 4870   × cxp 5653  Lim wlim 6378  ‘cfv 6555  rankcrnk 9791

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-reg 9610  ax-inf2 9666

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792  df-rank 9793

This theorem is used by:  rankxpsuc  9923