Theorem rankxplim2 8354

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 5502	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> (/) e. ( rank ` ( A X. B ) ) )
2		n0i 3767	. . . 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 2621	. . . 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 6607	. . . . . 6 |- ( A X. B ) e. _V
8	7	rankeq0 8335	. . . . 5 |- ( ( A X. B ) = (/) <-> ( rank ` ( A X. B ) ) = (/) )
9	8	notbii 298	. . . 4 |- ( -. ( A X. B ) = (/) <-> -. ( rank ` ( A X. B ) ) = (/) )
10	4, 9	bitr2i 254	. . 3 |- ( -. ( rank ` ( A X. B ) ) = (/) <-> ( A X. B ) =/= (/) )
11	3, 10	sylib 200	. 2 |- ( Lim ( rank ` ( A X. B ) ) -> ( A X. B ) =/= (/) )
12		limuni2 5501	. . . 4 |- ( Lim ( rank ` ( A X. B ) ) -> Lim U. ( rank ` ( A X. B ) ) )
13		limuni2 5501	. . . 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 8337	. . . . . 6 |- ( rank ` U. U. ( A X. B ) ) = U. ( rank ` U. ( A X. B ) )
16		rankuni 8337	. . . . . . 7 |- ( rank ` U. ( A X. B ) ) = U. ( rank ` ( A X. B ) )
17	16	unieqi 4226	. . . . . 6 |- U. ( rank ` U. ( A X. B ) ) = U. U. ( rank ` ( A X. B ) )
18	15, 17	eqtr2i 2453	. . . . 5 |- U. U. ( rank ` ( A X. B ) ) = ( rank ` U. U. ( A X. B ) )
19		unixp 5386	. . . . . 6 |- ( ( A X. B ) =/= (/) -> U. U. ( A X. B ) = ( A u. B ) )
20	19	fveq2d 5883	. . . . 5 |- ( ( A X. B ) =/= (/) -> ( rank ` U. U. ( A X. B ) ) = ( rank ` ( A u. B ) ) )
21	18, 20	syl5eq 2476	. . . 4 |- ( ( A X. B ) =/= (/) -> U. U. ( rank ` ( A X. B ) ) = ( rank ` ( A u. B ) ) )
22		limeq 5452	. . . 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 223	. 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 188   = wceq 1438   e. wcel 1869   =/= wne 2619   _Vcvv 3082   u. cun 3435   (/)c0 3762   U.cuni 4217   X. cxp 4849   Lim wlim 5441   ` cfv 5599   rankcrnk 8237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-reg 8111  ax-inf2 8150

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-om 6705  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-r1 8238  df-rank 8239

This theorem is referenced by:  rankxpsuc  8356