Theorem rankxplim2 8743

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 5787	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> (/) e. ( rank ` ( A X. B ) ) )
2		n0i 3920	. . . 4 |- ( (/) e. ( rank ` ( A X. B ) ) -> -. ( rank ` ( A X. B ) ) = (/) )
3	1, 2	syl 17	. . 3 |- ( Lim ( rank ` ( A X. B ) ) -> -. ( rank ` ( A X. B ) ) = (/) )
4		df-ne 2795	. . . 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 6962	. . . . . 6 |- ( A X. B ) e. _V
8	7	rankeq0 8724	. . . . 5 |- ( ( A X. B ) = (/) <-> ( rank ` ( A X. B ) ) = (/) )
9	8	notbii 310	. . . 4 |- ( -. ( A X. B ) = (/) <-> -. ( rank ` ( A X. B ) ) = (/) )
10	4, 9	bitr2i 265	. . 3 |- ( -. ( rank ` ( A X. B ) ) = (/) <-> ( A X. B ) =/= (/) )
11	3, 10	sylib 208	. 2 |- ( Lim ( rank ` ( A X. B ) ) -> ( A X. B ) =/= (/) )
12		limuni2 5786	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> Lim U. ( rank ` ( A X. B ) ) )
13		limuni2 5786	. . . 4 |- ( Lim U. ( rank ` ( A X. B ) ) -> Lim U. U. ( rank ` ( A X. B ) ) )
14	12, 13	syl 17	. . 3 |- ( Lim ( rank ` ( A X. B ) ) -> Lim U. U. ( rank ` ( A X. B ) ) )
15		rankuni 8726	. . . . . 6 |- ( rank ` U. U. ( A X. B ) ) = U. ( rank ` U. ( A X. B ) )
16		rankuni 8726	. . . . . . 7 |- ( rank ` U. ( A X. B ) ) = U. ( rank ` ( A X. B ) )
17	16	unieqi 4445	. . . . . 6 |- U. ( rank ` U. ( A X. B ) ) = U. U. ( rank ` ( A X. B ) )
18	15, 17	eqtr2i 2645	. . . . 5 |- U. U. ( rank ` ( A X. B ) ) = ( rank ` U. U. ( A X. B ) )
19		unixp 5668	. . . . . 6 |- ( ( A X. B ) =/= (/) -> U. U. ( A X. B ) = ( A u. B ) )
20	19	fveq2d 6195	. . . . 5 |- ( ( A X. B ) =/= (/) -> ( rank ` U. U. ( A X. B ) ) = ( rank ` ( A u. B ) ) )
21	18, 20	syl5eq 2668	. . . 4 |- ( ( A X. B ) =/= (/) -> U. U. ( rank ` ( A X. B ) ) = ( rank ` ( A u. B ) ) )
22		limeq 5735	. . . 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 17	. . 3 |- ( ( A X. B ) =/= (/) -> ( Lim U. U. ( rank ` ( A X. B ) ) <-> Lim ( rank ` ( A u. B ) ) ) )
24	14, 23	syl5ib 234	. 2 |- ( ( A X. B ) =/= (/) -> ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) ) )
25	11, 24	mpcom 38	1 |- ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   (/)c0 3915   U.cuni 4436   X. cxp 5112   Lim wlim 5724   ` cfv 5888   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  rankxpsuc  8745