Theorem ltfinirr 4457

Description: Irreflexive law for finite less than. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
ltfinirr	|- (A e. Nn -> -. <<A, A>> e. <fin )

Proof of Theorem ltfinirr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cnsuc 4401	. . . . . . . 8 |- (x +o 1c) =/= 0c
2	1	necomi 2598	. . . . . . 7 |- 0c =/= (x +o 1c)
3		df-ne 2518	. . . . . . 7 |- (0c =/= (x +o 1c) <-> -. 0c = (x +o 1c))
4	2, 3	mpbi 199	. . . . . 6 |- -. 0c = (x +o 1c)
5		addcid1 4405	. . . . . . . . 9 |- (A +o 0c) = A
6	5	eqcomi 2357	. . . . . . . 8 |- A = (A +o 0c)
7		addcass 4415	. . . . . . . 8 |- ((A +o x) +o 1c) = (A +o (x +o 1c))
8	6, 7	eqeq12i 2366	. . . . . . 7 |- (A = ((A +o x) +o 1c) <-> (A +o 0c) = (A +o (x +o 1c)))
9		simpll 730	. . . . . . . 8 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> A e. Nn )
10		peano1 4402	. . . . . . . . 9 |- 0c e. Nn
11	10	a1i 10	. . . . . . . 8 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> 0c e. Nn )
12		peano2 4403	. . . . . . . . 9 |- (x e. Nn -> (x +o 1c) e. Nn )
13	12	adantl 452	. . . . . . . 8 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> (x +o 1c) e. Nn )
14	5	neeq1i 2526	. . . . . . . . . 10 |- ((A +o 0c) =/= (/) <-> A =/= (/))
15	14	biimpri 197	. . . . . . . . 9 |- (A =/= (/) -> (A +o 0c) =/= (/))
16	15	ad2antlr 707	. . . . . . . 8 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> (A +o 0c) =/= (/))
17		preaddccan2 4455	. . . . . . . 8 |- (((A e. Nn /\ 0c e. Nn /\ (x +o 1c) e. Nn ) /\ (A +o 0c) =/= (/)) -> ((A +o 0c) = (A +o (x +o 1c)) <-> 0c = (x +o 1c)))
18	9, 11, 13, 16, 17	syl31anc 1185	. . . . . . 7 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> ((A +o 0c) = (A +o (x +o 1c)) <-> 0c = (x +o 1c)))
19	8, 18	syl5bb 248	. . . . . 6 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> (A = ((A +o x) +o 1c) <-> 0c = (x +o 1c)))
20	4, 19	mtbiri 294	. . . . 5 |- (((A e. Nn /\ A =/= (/)) /\ x e. Nn ) -> -. A = ((A +o x) +o 1c))
21	20	nrexdv 2717	. . . 4 |- ((A e. Nn /\ A =/= (/)) -> -. E.x e. Nn A = ((A +o x) +o 1c))
22	21	ex 423	. . 3 |- (A e. Nn -> (A =/= (/) -> -. E.x e. Nn A = ((A +o x) +o 1c)))
23		imnan 411	. . 3 |- ((A =/= (/) -> -. E.x e. Nn A = ((A +o x) +o 1c)) <-> -. (A =/= (/) /\ E.x e. Nn A = ((A +o x) +o 1c)))
24	22, 23	sylib 188	. 2 |- (A e. Nn -> -. (A =/= (/) /\ E.x e. Nn A = ((A +o x) +o 1c)))
25		opkltfing 4449	. . 3 |- ((A e. Nn /\ A e. Nn ) -> (<<A, A>> e. <fin <-> (A =/= (/) /\ E.x e. Nn A = ((A +o x) +o 1c))))
26	25	anidms 626	. 2 |- (A e. Nn -> (<<A, A>> e. <fin <-> (A =/= (/) /\ E.x e. Nn A = ((A +o x) +o 1c))))
27	24, 26	mtbird 292	1 |- (A e. Nn -> -. <<A, A>> e. <fin )

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = 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 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-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-ltfin 4441

This theorem is referenced by:  ltfinasym  4460  lenltfin  4469  tfinltfin  4501  sfin111  4536  vfinncvntnn  4548