Theorem ltfinp1 4463

Description: One plus a finite cardinal is strictly greater. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
ltfinp1	|- ((A e. V /\ A =/= (/)) -> <<A, (A +o 1c)>> e. <fin )

Proof of Theorem ltfinp1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 447	. . 3 |- ((A e. V /\ A =/= (/)) -> A =/= (/))
2		peano1 4403	. . . 4 |- 0c e. Nn
3		addcid1 4406	. . . . . 6 |- (A +o 0c) = A
4	3	addceq1i 4387	. . . . 5 |- ((A +o 0c) +o 1c) = (A +o 1c)
5	4	eqcomi 2357	. . . 4 |- (A +o 1c) = ((A +o 0c) +o 1c)
6		addceq2 4385	. . . . . . 7 |- (x = 0c -> (A +o x) = (A +o 0c))
7	6	addceq1d 4390	. . . . . 6 |- (x = 0c -> ((A +o x) +o 1c) = ((A +o 0c) +o 1c))
8	7	eqeq2d 2364	. . . . 5 |- (x = 0c -> ((A +o 1c) = ((A +o x) +o 1c) <-> (A +o 1c) = ((A +o 0c) +o 1c)))
9	8	rspcev 2956	. . . 4 |- ((0c e. Nn /\ (A +o 1c) = ((A +o 0c) +o 1c)) -> E.x e. Nn (A +o 1c) = ((A +o x) +o 1c))
10	2, 5, 9	mp2an 653	. . 3 |- E.x e. Nn (A +o 1c) = ((A +o x) +o 1c)
11	1, 10	jctir 524	. 2 |- ((A e. V /\ A =/= (/)) -> (A =/= (/) /\ E.x e. Nn (A +o 1c) = ((A +o x) +o 1c)))
12		1cex 4143	. . . . 5 |- 1c e. _V
13		addcexg 4394	. . . . 5 |- ((A e. V /\ 1c e. _V) -> (A +o 1c) e. _V)
14	12, 13	mpan2 652	. . . 4 |- (A e. V -> (A +o 1c) e. _V)
15		opkltfing 4450	. . . 4 |- ((A e. V /\ (A +o 1c) e. _V) -> (<<A, (A +o 1c)>> e. <fin <-> (A =/= (/) /\ E.x e. Nn (A +o 1c) = ((A +o x) +o 1c))))
16	14, 15	mpdan 649	. . 3 |- (A e. V -> (<<A, (A +o 1c)>> e. <fin <-> (A =/= (/) /\ E.x e. Nn (A +o 1c) = ((A +o x) +o 1c))))
17	16	adantr 451	. 2 |- ((A e. V /\ A =/= (/)) -> (<<A, (A +o 1c)>> e. <fin <-> (A =/= (/) /\ E.x e. Nn (A +o 1c) = ((A +o x) +o 1c))))
18	11, 17	mpbird 223	1 |- ((A e. V /\ A =/= (/)) -> <<A, (A +o 1c)>> e. <fin )

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  _Vcvv 2860  (/)c0 3551  <<copk 4058  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   <_fin cltfin 4434

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194  df-0c 4378  df-addc 4379  df-nnc 4380  df-ltfin 4442

This theorem is referenced by:  ltfintri  4467