Theorem rankxplim2 8248

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 4881	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> (/) e. ( rank ` ( A X. B ) ) )
2		n0i 3740	. . . 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 2598	. . . 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 6540	. . . . . 6 |- ( A X. B ) e. _V
8	7	rankeq0 8229	. . . . 5 |- ( ( A X. B ) = (/) <-> ( rank ` ( A X. B ) ) = (/) )
9	8	notbii 294	. . . 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 4880	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> Lim U. ( rank ` ( A X. B ) ) )
13		limuni2 4880	. . . 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 8231	. . . . . 6 |- ( rank ` U. U. ( A X. B ) ) = U. ( rank ` U. ( A X. B ) )
16		rankuni 8231	. . . . . . 7 |- ( rank ` U. ( A X. B ) ) = U. ( rank ` ( A X. B ) )
17	16	unieqi 4197	. . . . . 6 |- U. ( rank ` U. ( A X. B ) ) = U. U. ( rank ` ( A X. B ) )
18	15, 17	eqtr2i 2430	. . . . 5 |- U. U. ( rank ` ( A X. B ) ) = ( rank ` U. U. ( A X. B ) )
19		unixp 5476	. . . . . 6 |- ( ( A X. B ) =/= (/) -> U. U. ( A X. B ) = ( A u. B ) )
20	19	fveq2d 5807	. . . . 5 |- ( ( A X. B ) =/= (/) -> ( rank ` U. U. ( A X. B ) ) = ( rank ` ( A u. B ) ) )
21	18, 20	syl5eq 2453	. . . 4 |- ( ( A X. B ) =/= (/) -> U. U. ( rank ` ( A X. B ) ) = ( rank ` ( A u. B ) ) )
22		limeq 4831	. . . 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 219	. 2 |- ( ( A X. B ) =/= (/) -> ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) ) )
25	11, 24	mpcom 34	1 |- ( Lim ( rank ` ( A X. B ) ) -> Lim ( rank ` ( A u. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   = wceq 1403   e. wcel 1840   =/= wne 2596   _Vcvv 3056   u. cun 3409   (/)c0 3735   U.cuni 4188   Lim wlim 4820   X. cxp 4938   ` cfv 5523   rankcrnk 8131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-reg 7970  ax-inf2 8009

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-om 6637  df-recs 6997  df-rdg 7031  df-r1 8132  df-rank 8133

This theorem is referenced by:  rankxpsuc  8250