Theorem rankxplim2 8201

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 e. _V
rankxplim.2	|- B e. _V

Assertion

Ref	Expression
rankxplim2	|- ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) )

Proof of Theorem rankxplim2

Step	Hyp	Ref	Expression
1		0ellim 4892	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> (/) e. ( rank ` ( A X. B ) ) )
2		n0i 3753	. . . 4 |- ( (/) e. ( rank ` ( A X. B ) ) -> -. ( rank ` ( A X. B ) ) = (/) )
3	1, 2	syl 16	. . 3 |- ( Lim ( rank ` ( A X. B ) ) -> -. ( rank ` ( A X. B ) ) = (/) )
4		df-ne 2650	. . . 4 |- ( ( A X. B ) =/= (/) <-> -. ( A X. B ) = (/) )
5		rankxplim.1	. . . . . . 7 |- A e. _V
6		rankxplim.2	. . . . . . 7 |- B e. _V
7	5, 6	xpex 6621	. . . . . 6 |- ( A X. B ) e. _V
8	7	rankeq0 8182	. . . . 5 |- ( ( A X. B ) = (/) <-> ( rank ` ( A X. B ) ) = (/) )
9	8	notbii 296	. . . 4 |- ( -. ( A X. B ) = (/) <-> -. ( rank ` ( A X. B ) ) = (/) )
10	4, 9	bitr2i 250	. . 3 |- ( -. ( rank ` ( A X. B ) ) = (/) <-> ( A X. B ) =/= (/) )
11	3, 10	sylib 196	. 2 |- ( Lim ( rank ` ( A X. B ) ) -> ( A X. B ) =/= (/) )
12		limuni2 4891	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> Lim U. ( rank ` ( A X. B ) ) )
13		limuni2 4891	. . . 4 |- ( Lim U. ( rank ` ( A X. B ) ) -> Lim U. U. ( rank ` ( A X. B ) ) )
14	12, 13	syl 16	. . 3 |- ( Lim ( rank ` ( A X. B ) ) -> Lim U. U. ( rank ` ( A X. B ) ) )
15		rankuni 8184	. . . . . 6 |- ( rank ` U. U. ( A X. B ) ) = U. ( rank ` U. ( A X. B ) )
16		rankuni 8184	. . . . . . 7 |- ( rank ` U. ( A X. B ) ) = U. ( rank ` ( A X. B ) )
17	16	unieqi 4211	. . . . . 6 |- U. ( rank ` U. ( A X. B ) ) = U. U. ( rank ` ( A X. B ) )
18	15, 17	eqtr2i 2484	. . . . 5 |- U. U. ( rank ` ( A X. B ) ) = ( rank ` U. U. ( A X. B ) )
19		unixp 5481	. . . . . 6 |- ( ( A X. B ) =/= (/) -> U. U. ( A X. B ) = ( A u. B ) )
20	19	fveq2d 5806	. . . . 5 |- ( ( A X. B ) =/= (/) -> ( rank ` U. U. ( A X. B ) ) = ( rank ` ( A u. B ) ) )
21	18, 20	syl5eq 2507	. . . 4 |- ( ( A X. B ) =/= (/) -> U. U. ( rank ` ( A X. B ) ) = ( rank ` ( A u. B ) ) )
22		limeq 4842	. . . 4 |- ( U. U. ( rank ` ( A X. B ) ) = ( rank ` ( A u. B ) ) -> ( Lim U. U. ( rank ` ( A X. B ) ) <-> Lim ( rank ` ( A u. B ) ) ) )
23	21, 22	syl 16	. . 3 |- ( ( A X. B ) =/= (/) -> ( Lim U. U. ( rank ` ( A X. B ) ) <-> Lim ( rank ` ( A u. B ) ) ) )
24	14, 23	syl5ib 219	. 2 |- ( ( A X. B ) =/= (/) -> ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) ) )
25	11, 24	mpcom 36	1 |- ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   = wceq 1370   e. wcel 1758   =/= wne 2648   _Vcvv 3078   u. cun 3437   (/)c0 3748   U.cuni 4202   Lim wlim 4831   X. cxp 4949   ` cfv 5529   rankcrnk 8084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-inf2 7961

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979  df-r1 8085  df-rank 8086

This theorem is referenced by:  rankxpsuc  8203