Theorem lefinlteq 4463

Description: Transfer from less than or equal to less than. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
lefinlteq	|- ((A e. V /\ B e. W /\ A =/= (/)) -> (<<A, B>> e. <_fin <-> (<<A, B>> e. <fin \/ A = B)))

Proof of Theorem lefinlteq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnc0suc 4412	. . . . . . . 8 |- (x e. Nn <-> (x = 0c \/ E.y e. Nn x = (y +o 1c)))
2		addceq2 4384	. . . . . . . . . 10 |- (x = 0c -> (A +o x) = (A +o 0c))
3		addcid1 4405	. . . . . . . . . 10 |- (A +o 0c) = A
4	2, 3	syl6req 2402	. . . . . . . . 9 |- (x = 0c -> A = (A +o x))
5		addceq2 4384	. . . . . . . . . . 11 |- (x = (y +o 1c) -> (A +o x) = (A +o (y +o 1c)))
6		addcass 4415	. . . . . . . . . . 11 |- ((A +o y) +o 1c) = (A +o (y +o 1c))
7	5, 6	syl6eqr 2403	. . . . . . . . . 10 |- (x = (y +o 1c) -> (A +o x) = ((A +o y) +o 1c))
8	7	reximi 2721	. . . . . . . . 9 |- (E.y e. Nn x = (y +o 1c) -> E.y e. Nn (A +o x) = ((A +o y) +o 1c))
9	4, 8	orim12i 502	. . . . . . . 8 |- ((x = 0c \/ E.y e. Nn x = (y +o 1c)) -> (A = (A +o x) \/ E.y e. Nn (A +o x) = ((A +o y) +o 1c)))
10	1, 9	sylbi 187	. . . . . . 7 |- (x e. Nn -> (A = (A +o x) \/ E.y e. Nn (A +o x) = ((A +o y) +o 1c)))
11	10	orcomd 377	. . . . . 6 |- (x e. Nn -> (E.y e. Nn (A +o x) = ((A +o y) +o 1c) \/ A = (A +o x)))
12		eqeq1 2359	. . . . . . . 8 |- (B = (A +o x) -> (B = ((A +o y) +o 1c) <-> (A +o x) = ((A +o y) +o 1c)))
13	12	rexbidv 2635	. . . . . . 7 |- (B = (A +o x) -> (E.y e. Nn B = ((A +o y) +o 1c) <-> E.y e. Nn (A +o x) = ((A +o y) +o 1c)))
14		eqeq2 2362	. . . . . . 7 |- (B = (A +o x) -> (A = B <-> A = (A +o x)))
15	13, 14	orbi12d 690	. . . . . 6 |- (B = (A +o x) -> ((E.y e. Nn B = ((A +o y) +o 1c) \/ A = B) <-> (E.y e. Nn (A +o x) = ((A +o y) +o 1c) \/ A = (A +o x))))
16	11, 15	syl5ibrcom 213	. . . . 5 |- (x e. Nn -> (B = (A +o x) -> (E.y e. Nn B = ((A +o y) +o 1c) \/ A = B)))
17	16	rexlimiv 2732	. . . 4 |- (E.x e. Nn B = (A +o x) -> (E.y e. Nn B = ((A +o y) +o 1c) \/ A = B))
18	6	eqeq2i 2363	. . . . . . 7 |- (B = ((A +o y) +o 1c) <-> B = (A +o (y +o 1c)))
19		peano2 4403	. . . . . . . 8 |- (y e. Nn -> (y +o 1c) e. Nn )
20	5	eqeq2d 2364	. . . . . . . . 9 |- (x = (y +o 1c) -> (B = (A +o x) <-> B = (A +o (y +o 1c))))
21	20	rspcev 2955	. . . . . . . 8 |- (((y +o 1c) e. Nn /\ B = (A +o (y +o 1c))) -> E.x e. Nn B = (A +o x))
22	19, 21	sylan 457	. . . . . . 7 |- ((y e. Nn /\ B = (A +o (y +o 1c))) -> E.x e. Nn B = (A +o x))
23	18, 22	sylan2b 461	. . . . . 6 |- ((y e. Nn /\ B = ((A +o y) +o 1c)) -> E.x e. Nn B = (A +o x))
24	23	rexlimiva 2733	. . . . 5 |- (E.y e. Nn B = ((A +o y) +o 1c) -> E.x e. Nn B = (A +o x))
25		peano1 4402	. . . . . . 7 |- 0c e. Nn
26	3	eqcomi 2357	. . . . . . 7 |- A = (A +o 0c)
27	2	eqeq2d 2364	. . . . . . . 8 |- (x = 0c -> (A = (A +o x) <-> A = (A +o 0c)))
28	27	rspcev 2955	. . . . . . 7 |- ((0c e. Nn /\ A = (A +o 0c)) -> E.x e. Nn A = (A +o x))
29	25, 26, 28	mp2an 653	. . . . . 6 |- E.x e. Nn A = (A +o x)
30		eqeq1 2359	. . . . . . 7 |- (A = B -> (A = (A +o x) <-> B = (A +o x)))
31	30	rexbidv 2635	. . . . . 6 |- (A = B -> (E.x e. Nn A = (A +o x) <-> E.x e. Nn B = (A +o x)))
32	29, 31	mpbii 202	. . . . 5 |- (A = B -> E.x e. Nn B = (A +o x))
33	24, 32	jaoi 368	. . . 4 |- ((E.y e. Nn B = ((A +o y) +o 1c) \/ A = B) -> E.x e. Nn B = (A +o x))
34	17, 33	impbii 180	. . 3 |- (E.x e. Nn B = (A +o x) <-> (E.y e. Nn B = ((A +o y) +o 1c) \/ A = B))
35	34	a1i 10	. 2 |- ((A e. V /\ B e. W /\ A =/= (/)) -> (E.x e. Nn B = (A +o x) <-> (E.y e. Nn B = ((A +o y) +o 1c) \/ A = B)))
36		opklefing 4448	. . 3 |- ((A e. V /\ B e. W) -> (<<A, B>> e. <_fin <-> E.x e. Nn B = (A +o x)))
37	36	3adant3 975	. 2 |- ((A e. V /\ B e. W /\ A =/= (/)) -> (<<A, B>> e. <_fin <-> E.x e. Nn B = (A +o x)))
38		opkltfing 4449	. . . . . 6 |- ((A e. V /\ B e. W) -> (<<A, B>> e. <fin <-> (A =/= (/) /\ E.y e. Nn B = ((A +o y) +o 1c))))
39	38	adantr 451	. . . . 5 |- (((A e. V /\ B e. W) /\ A =/= (/)) -> (<<A, B>> e. <fin <-> (A =/= (/) /\ E.y e. Nn B = ((A +o y) +o 1c))))
40		ibar 490	. . . . . 6 |- (A =/= (/) -> (E.y e. Nn B = ((A +o y) +o 1c) <-> (A =/= (/) /\ E.y e. Nn B = ((A +o y) +o 1c))))
41	40	adantl 452	. . . . 5 |- (((A e. V /\ B e. W) /\ A =/= (/)) -> (E.y e. Nn B = ((A +o y) +o 1c) <-> (A =/= (/) /\ E.y e. Nn B = ((A +o y) +o 1c))))
42	39, 41	bitr4d 247	. . . 4 |- (((A e. V /\ B e. W) /\ A =/= (/)) -> (<<A, B>> e. <fin <-> E.y e. Nn B = ((A +o y) +o 1c)))
43	42	orbi1d 683	. . 3 |- (((A e. V /\ B e. W) /\ A =/= (/)) -> ((<<A, B>> e. <fin \/ A = B) <-> (E.y e. Nn B = ((A +o y) +o 1c) \/ A = B)))
44	43	3impa 1146	. 2 |- ((A e. V /\ B e. W /\ A =/= (/)) -> ((<<A, B>> e. <fin \/ A = B) <-> (E.y e. Nn B = ((A +o y) +o 1c) \/ A = B)))
45	35, 37, 44	3bitr4d 276	1 |- ((A e. V /\ B e. W /\ A =/= (/)) -> (<<A, B>> e. <_fin <-> (<<A, B>> e. <fin \/ A = B)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2516  E.wrex 2615  (/)c0 3550  <<copk 4057  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +o cplc 4375   <__fin clefin 4432   <_fin cltfin 4433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-0c 4377  df-addc 4378  df-nnc 4379  df-lefin 4440  df-ltfin 4441

This theorem is referenced by:  ltfintri  4466  vfin1cltv  4547