Theorem genpn0 9825

Description: The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
genp.1	|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )
genp.2	|- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )

Assertion

Ref	Expression
genpn0	|- ( ( A e. P. /\ B e. P. ) -> (/) C. ( A F B ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, w, v, G, y, z

Allowed substitution hints:   A( w, v)   B( w, v)   F( x, y, z, w, v)

Proof of Theorem genpn0

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prn0 9811	. . . 4 |- ( A e. P. -> A =/= (/) )
2		n0 3931	. . . 4 |- ( A =/= (/) <-> E. f f e. A )
3	1, 2	sylib 208	. . 3 |- ( A e. P. -> E. f f e. A )
4		prn0 9811	. . . 4 |- ( B e. P. -> B =/= (/) )
5		n0 3931	. . . 4 |- ( B =/= (/) <-> E. g g e. B )
6	4, 5	sylib 208	. . 3 |- ( B e. P. -> E. g g e. B )
7	3, 6	anim12i 590	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( E. f f e. A /\ E. g g e. B ) )
8		genp.1	. . . . . . . . 9 |- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )
9		genp.2	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
10	8, 9	genpprecl 9823	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( ( f e. A /\ g e. B ) -> ( f G g ) e. ( A F B ) ) )
11		ne0i 3921	. . . . . . . . 9 |- ( ( f G g ) e. ( A F B ) -> ( A F B ) =/= (/) )
12		0pss 4013	. . . . . . . . 9 |- ( (/) C. ( A F B ) <-> ( A F B ) =/= (/) )
13	11, 12	sylibr 224	. . . . . . . 8 |- ( ( f G g ) e. ( A F B ) -> (/) C. ( A F B ) )
14	10, 13	syl6 35	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( ( f e. A /\ g e. B ) -> (/) C. ( A F B ) ) )
15	14	expcomd 454	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( g e. B -> ( f e. A -> (/) C. ( A F B ) ) ) )
16	15	exlimdv 1861	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( E. g g e. B -> ( f e. A -> (/) C. ( A F B ) ) ) )
17	16	com23 86	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( f e. A -> ( E. g g e. B -> (/) C. ( A F B ) ) ) )
18	17	exlimdv 1861	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( E. f f e. A -> ( E. g g e. B -> (/) C. ( A F B ) ) ) )
19	18	impd 447	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( ( E. f f e. A /\ E. g g e. B ) -> (/) C. ( A F B ) ) )
20	7, 19	mpd 15	1 |- ( ( A e. P. /\ B e. P. ) -> (/) C. ( A F B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   C. wpss 3575   (/)c0 3915  (class class class)co 6650   |-> cmpt2 6652   Q.cnq 9674   P.cnp 9681

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-ni 9694  df-nq 9734  df-np 9803

This theorem is referenced by:  genpcl  9830