Theorem vfin1cltv 4548

Description: If the universe is finite, then 1_c is strictly smaller than the universe. Theorem X.1.57 of [Rosser] p. 534. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
vfin1cltv	|- (_V e. Fin -> << Ncfin 1c, Ncfin _V>> e. <fin )

Proof of Theorem vfin1cltv

Step	Hyp	Ref	Expression
1		uncompl 4075	. . . . 5 |- (1c u. ∼ 1c) = _V
2		ncfineq 4474	. . . . 5 |- ((1c u. ∼ 1c) = _V -> Ncfin (1c u. ∼ 1c) = Ncfin _V)
3	1, 2	ax-mp 5	. . . 4 |- Ncfin (1c u. ∼ 1c) = Ncfin _V
4		1cex 4143	. . . . 5 |- 1c e. _V
5	4	complex 4105	. . . . . 6 |- ∼ 1c e. _V
6		incompl 4074	. . . . . 6 |- (1c i^i ∼ 1c) = (/)
7		ncfindi 4476	. . . . . 6 |- (((_V e. Fin /\ 1c e. _V) /\ ∼ 1c e. _V /\ (1c i^i ∼ 1c) = (/)) -> Ncfin (1c u. ∼ 1c) = ( Ncfin 1c +o Ncfin ∼ 1c))
8	5, 6, 7	mp3an23 1269	. . . . 5 |- ((_V e. Fin /\ 1c e. _V) -> Ncfin (1c u. ∼ 1c) = ( Ncfin 1c +o Ncfin ∼ 1c))
9	4, 8	mpan2 652	. . . 4 |- (_V e. Fin -> Ncfin (1c u. ∼ 1c) = ( Ncfin 1c +o Ncfin ∼ 1c))
10	3, 9	syl5reqr 2400	. . 3 |- (_V e. Fin -> ( Ncfin 1c +o Ncfin ∼ 1c) = Ncfin _V)
11	10	opkeq2d 4067	. 2 |- (_V e. Fin -> << Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> = << Ncfin 1c, Ncfin _V>>)
12		0nel1c 4160	. . . . . . 7 |- -. (/) e. 1c
13		0ex 4111	. . . . . . . 8 |- (/) e. _V
14	13	elcompl 3226	. . . . . . 7 |- ((/) e. ∼ 1c <-> -. (/) e. 1c)
15	12, 14	mpbir 200	. . . . . 6 |- (/) e. ∼ 1c
16		n0i 3556	. . . . . 6 |- ((/) e. ∼ 1c -> -. ∼ 1c = (/))
17	15, 16	ax-mp 5	. . . . 5 |- -. ∼ 1c = (/)
18		ncfinprop 4475	. . . . . . . . 9 |- ((_V e. Fin /\ ∼ 1c e. _V) -> ( Ncfin ∼ 1c e. Nn /\ ∼ 1c e. Ncfin ∼ 1c))
19	5, 18	mpan2 652	. . . . . . . 8 |- (_V e. Fin -> ( Ncfin ∼ 1c e. Nn /\ ∼ 1c e. Ncfin ∼ 1c))
20	19	simprd 449	. . . . . . 7 |- (_V e. Fin -> ∼ 1c e. Ncfin ∼ 1c)
21		eleq2 2414	. . . . . . 7 |- (0c = Ncfin ∼ 1c -> ( ∼ 1c e. 0c <-> ∼ 1c e. Ncfin ∼ 1c))
22	20, 21	syl5ibrcom 213	. . . . . 6 |- (_V e. Fin -> (0c = Ncfin ∼ 1c -> ∼ 1c e. 0c))
23		el0c 4422	. . . . . 6 |- ( ∼ 1c e. 0c <-> ∼ 1c = (/))
24	22, 23	syl6ib 217	. . . . 5 |- (_V e. Fin -> (0c = Ncfin ∼ 1c -> ∼ 1c = (/)))
25	17, 24	mtoi 169	. . . 4 |- (_V e. Fin -> -. 0c = Ncfin ∼ 1c)
26		addcid1 4406	. . . . . 6 |- ( Ncfin 1c +o 0c) = Ncfin 1c
27	26	eqeq1i 2360	. . . . 5 |- (( Ncfin 1c +o 0c) = ( Ncfin 1c +o Ncfin ∼ 1c) <-> Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c))
28		ncfinprop 4475	. . . . . . . 8 |- ((_V e. Fin /\ 1c e. _V) -> ( Ncfin 1c e. Nn /\ 1c e. Ncfin 1c))
29	4, 28	mpan2 652	. . . . . . 7 |- (_V e. Fin -> ( Ncfin 1c e. Nn /\ 1c e. Ncfin 1c))
30	29	simpld 445	. . . . . 6 |- (_V e. Fin -> Ncfin 1c e. Nn )
31		peano1 4403	. . . . . . 7 |- 0c e. Nn
32	31	a1i 10	. . . . . 6 |- (_V e. Fin -> 0c e. Nn )
33	19	simpld 445	. . . . . 6 |- (_V e. Fin -> Ncfin ∼ 1c e. Nn )
34	26	a1i 10	. . . . . . 7 |- (_V e. Fin -> ( Ncfin 1c +o 0c) = Ncfin 1c)
35	29	simprd 449	. . . . . . . 8 |- (_V e. Fin -> 1c e. Ncfin 1c)
36		ne0i 3557	. . . . . . . 8 |- (1c e. Ncfin 1c -> Ncfin 1c =/= (/))
37	35, 36	syl 15	. . . . . . 7 |- (_V e. Fin -> Ncfin 1c =/= (/))
38	34, 37	eqnetrd 2535	. . . . . 6 |- (_V e. Fin -> ( Ncfin 1c +o 0c) =/= (/))
39		preaddccan2 4456	. . . . . 6 |- ((( Ncfin 1c e. Nn /\ 0c e. Nn /\ Ncfin ∼ 1c e. Nn ) /\ ( Ncfin 1c +o 0c) =/= (/)) -> (( Ncfin 1c +o 0c) = ( Ncfin 1c +o Ncfin ∼ 1c) <-> 0c = Ncfin ∼ 1c))
40	30, 32, 33, 38, 39	syl31anc 1185	. . . . 5 |- (_V e. Fin -> (( Ncfin 1c +o 0c) = ( Ncfin 1c +o Ncfin ∼ 1c) <-> 0c = Ncfin ∼ 1c))
41	27, 40	syl5bbr 250	. . . 4 |- (_V e. Fin -> ( Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c) <-> 0c = Ncfin ∼ 1c))
42	25, 41	mtbird 292	. . 3 |- (_V e. Fin -> -. Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c))
43		ncfinex 4473	. . . . . . 7 |- Ncfin 1c e. _V
44		lefinaddc 4451	. . . . . . 7 |- (( Ncfin 1c e. _V /\ Ncfin ∼ 1c e. Nn ) -> << Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <_fin )
45	43, 33, 44	sylancr 644	. . . . . 6 |- (_V e. Fin -> << Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <_fin )
46		ncfinex 4473	. . . . . . . . 9 |- Ncfin ∼ 1c e. _V
47	43, 46	addcex 4395	. . . . . . . 8 |- ( Ncfin 1c +o Ncfin ∼ 1c) e. _V
48		lefinlteq 4464	. . . . . . . 8 |- (( Ncfin 1c e. _V /\ ( Ncfin 1c +o Ncfin ∼ 1c) e. _V /\ Ncfin 1c =/= (/)) -> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <_fin <-> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin \/ Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c))))
49	43, 47, 48	mp3an12 1267	. . . . . . 7 |- ( Ncfin 1c =/= (/) -> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <_fin <-> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin \/ Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c))))
50	37, 49	syl 15	. . . . . 6 |- (_V e. Fin -> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <_fin <-> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin \/ Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c))))
51	45, 50	mpbid 201	. . . . 5 |- (_V e. Fin -> (<< Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin \/ Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c)))
52	51	orcomd 377	. . . 4 |- (_V e. Fin -> ( Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c) \/ << Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin ))
53	52	ord 366	. . 3 |- (_V e. Fin -> (-. Ncfin 1c = ( Ncfin 1c +o Ncfin ∼ 1c) -> << Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin ))
54	42, 53	mpd 14	. 2 |- (_V e. Fin -> << Ncfin 1c, ( Ncfin 1c +o Ncfin ∼ 1c)>> e. <fin )
55	11, 54	eqeltrrd 2428	1 |- (_V e. Fin -> << Ncfin 1c, Ncfin _V>> e. <fin )

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860   ∼ ccompl 3206   u. cun 3208   i^i cin 3209  (/)c0 3551  <<copk 4058  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   Fin cfin 4377   <__fin clefin 4433   <_fin cltfin 4434   Nc_fin cncfin 4435

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-14 1714  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-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443

This theorem is referenced by:  vfinncvntnn  4549