Theorem evenoddnnnul 4515

Description: Every nonempty finite cardinal is either even or odd. Theorem X.1.35 of [Rosser] p. 529. (Contributed by SF, 20-Jan-2015.)

Assertion

Ref	Expression
evenoddnnnul	|- ( Evenfin u. Oddfin ) = ( Nn \ {(/)})

Proof of Theorem evenoddnnnul

Dummy variables x k n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evennn 4507	. . . . . 6 |- (x e. Evenfin -> x e. Nn )
2		evennnul 4509	. . . . . 6 |- (x e. Evenfin -> x =/= (/))
3		eldifsn 3840	. . . . . 6 |- (x e. ( Nn \ {(/)}) <-> (x e. Nn /\ x =/= (/)))
4	1, 2, 3	sylanbrc 645	. . . . 5 |- (x e. Evenfin -> x e. ( Nn \ {(/)}))
5	4	ssriv 3278	. . . 4 |- Evenfin C_ ( Nn \ {(/)})
6		oddnn 4508	. . . . . 6 |- (x e. Oddfin -> x e. Nn )
7		oddnnul 4510	. . . . . 6 |- (x e. Oddfin -> x =/= (/))
8	6, 7, 3	sylanbrc 645	. . . . 5 |- (x e. Oddfin -> x e. ( Nn \ {(/)}))
9	8	ssriv 3278	. . . 4 |- Oddfin C_ ( Nn \ {(/)})
10	5, 9	pm3.2i 441	. . 3 |- ( Evenfin C_ ( Nn \ {(/)}) /\ Oddfin C_ ( Nn \ {(/)}))
11		unss 3438	. . 3 |- (( Evenfin C_ ( Nn \ {(/)}) /\ Oddfin C_ ( Nn \ {(/)})) <-> ( Evenfin u. Oddfin ) C_ ( Nn \ {(/)}))
12	10, 11	mpbi 199	. 2 |- ( Evenfin u. Oddfin ) C_ ( Nn \ {(/)})
13		eldifsn 3840	. . . 4 |- (n e. ( Nn \ {(/)}) <-> (n e. Nn /\ n =/= (/)))
14		vex 2863	. . . . . . . . . . . 12 |- m e. _V
15	14	elsnc 3757	. . . . . . . . . . 11 |- (m e. {(/)} <-> m = (/))
16		df-ne 2519	. . . . . . . . . . . 12 |- (m =/= (/) <-> -. m = (/))
17	16	con2bii 322	. . . . . . . . . . 11 |- (m = (/) <-> -. m =/= (/))
18	15, 17	bitri 240	. . . . . . . . . 10 |- (m e. {(/)} <-> -. m =/= (/))
19	18	orbi1i 506	. . . . . . . . 9 |- ((m e. {(/)} \/ m e. ( Evenfin u. Oddfin )) <-> (-. m =/= (/) \/ m e. ( Evenfin u. Oddfin )))
20		elun 3221	. . . . . . . . 9 |- (m e. ({(/)} u. ( Evenfin u. Oddfin )) <-> (m e. {(/)} \/ m e. ( Evenfin u. Oddfin )))
21		imor 401	. . . . . . . . 9 |- ((m =/= (/) -> m e. ( Evenfin u. Oddfin )) <-> (-. m =/= (/) \/ m e. ( Evenfin u. Oddfin )))
22	19, 20, 21	3bitr4i 268	. . . . . . . 8 |- (m e. ({(/)} u. ( Evenfin u. Oddfin )) <-> (m =/= (/) -> m e. ( Evenfin u. Oddfin )))
23	22	eqabi 2465	. . . . . . 7 |- ({(/)} u. ( Evenfin u. Oddfin )) = {m | (m =/= (/) -> m e. ( Evenfin u. Oddfin ))}
24		snex 4112	. . . . . . . 8 |- {(/)} e. _V
25		evenfinex 4504	. . . . . . . . 9 |- Evenfin e. _V
26		oddfinex 4505	. . . . . . . . 9 |- Oddfin e. _V
27	25, 26	unex 4107	. . . . . . . 8 |- ( Evenfin u. Oddfin ) e. _V
28	24, 27	unex 4107	. . . . . . 7 |- ({(/)} u. ( Evenfin u. Oddfin )) e. _V
29	23, 28	eqeltrri 2424	. . . . . 6 |- {m | (m =/= (/) -> m e. ( Evenfin u. Oddfin ))} e. _V
30		neeq1 2525	. . . . . . 7 |- (m = 0c -> (m =/= (/) <-> 0c =/= (/)))
31		eleq1 2413	. . . . . . 7 |- (m = 0c -> (m e. ( Evenfin u. Oddfin ) <-> 0c e. ( Evenfin u. Oddfin )))
32	30, 31	imbi12d 311	. . . . . 6 |- (m = 0c -> ((m =/= (/) -> m e. ( Evenfin u. Oddfin )) <-> (0c =/= (/) -> 0c e. ( Evenfin u. Oddfin ))))
33		neeq1 2525	. . . . . . 7 |- (m = k -> (m =/= (/) <-> k =/= (/)))
34		eleq1 2413	. . . . . . 7 |- (m = k -> (m e. ( Evenfin u. Oddfin ) <-> k e. ( Evenfin u. Oddfin )))
35	33, 34	imbi12d 311	. . . . . 6 |- (m = k -> ((m =/= (/) -> m e. ( Evenfin u. Oddfin )) <-> (k =/= (/) -> k e. ( Evenfin u. Oddfin ))))
36		neeq1 2525	. . . . . . 7 |- (m = (k +o 1c) -> (m =/= (/) <-> (k +o 1c) =/= (/)))
37		eleq1 2413	. . . . . . 7 |- (m = (k +o 1c) -> (m e. ( Evenfin u. Oddfin ) <-> (k +o 1c) e. ( Evenfin u. Oddfin )))
38	36, 37	imbi12d 311	. . . . . 6 |- (m = (k +o 1c) -> ((m =/= (/) -> m e. ( Evenfin u. Oddfin )) <-> ((k +o 1c) =/= (/) -> (k +o 1c) e. ( Evenfin u. Oddfin ))))
39		neeq1 2525	. . . . . . 7 |- (m = n -> (m =/= (/) <-> n =/= (/)))
40		eleq1 2413	. . . . . . 7 |- (m = n -> (m e. ( Evenfin u. Oddfin ) <-> n e. ( Evenfin u. Oddfin )))
41	39, 40	imbi12d 311	. . . . . 6 |- (m = n -> ((m =/= (/) -> m e. ( Evenfin u. Oddfin )) <-> (n =/= (/) -> n e. ( Evenfin u. Oddfin ))))
42		ssun1 3427	. . . . . . . 8 |- Evenfin C_ ( Evenfin u. Oddfin )
43		0ceven 4506	. . . . . . . 8 |- 0c e. Evenfin
44	42, 43	sselii 3271	. . . . . . 7 |- 0c e. ( Evenfin u. Oddfin )
45	44	a1i 10	. . . . . 6 |- (0c =/= (/) -> 0c e. ( Evenfin u. Oddfin ))
46		addcnnul 4454	. . . . . . . . . 10 |- ((k +o 1c) =/= (/) -> (k =/= (/) /\ 1c =/= (/)))
47	46	simpld 445	. . . . . . . . 9 |- ((k +o 1c) =/= (/) -> k =/= (/))
48		sucevenodd 4511	. . . . . . . . . . . 12 |- ((k e. Evenfin /\ (k +o 1c) =/= (/)) -> (k +o 1c) e. Oddfin )
49	48	expcom 424	. . . . . . . . . . 11 |- ((k +o 1c) =/= (/) -> (k e. Evenfin -> (k +o 1c) e. Oddfin ))
50		sucoddeven 4512	. . . . . . . . . . . 12 |- ((k e. Oddfin /\ (k +o 1c) =/= (/)) -> (k +o 1c) e. Evenfin )
51	50	expcom 424	. . . . . . . . . . 11 |- ((k +o 1c) =/= (/) -> (k e. Oddfin -> (k +o 1c) e. Evenfin ))
52	49, 51	orim12d 811	. . . . . . . . . 10 |- ((k +o 1c) =/= (/) -> ((k e. Evenfin \/ k e. Oddfin ) -> ((k +o 1c) e. Oddfin \/ (k +o 1c) e. Evenfin )))
53		elun 3221	. . . . . . . . . 10 |- (k e. ( Evenfin u. Oddfin ) <-> (k e. Evenfin \/ k e. Oddfin ))
54		elun 3221	. . . . . . . . . . 11 |- ((k +o 1c) e. ( Evenfin u. Oddfin ) <-> ((k +o 1c) e. Evenfin \/ (k +o 1c) e. Oddfin ))
55		orcom 376	. . . . . . . . . . 11 |- (((k +o 1c) e. Evenfin \/ (k +o 1c) e. Oddfin ) <-> ((k +o 1c) e. Oddfin \/ (k +o 1c) e. Evenfin ))
56	54, 55	bitri 240	. . . . . . . . . 10 |- ((k +o 1c) e. ( Evenfin u. Oddfin ) <-> ((k +o 1c) e. Oddfin \/ (k +o 1c) e. Evenfin ))
57	52, 53, 56	3imtr4g 261	. . . . . . . . 9 |- ((k +o 1c) =/= (/) -> (k e. ( Evenfin u. Oddfin ) -> (k +o 1c) e. ( Evenfin u. Oddfin )))
58	47, 57	embantd 50	. . . . . . . 8 |- ((k +o 1c) =/= (/) -> ((k =/= (/) -> k e. ( Evenfin u. Oddfin )) -> (k +o 1c) e. ( Evenfin u. Oddfin )))
59	58	com12 27	. . . . . . 7 |- ((k =/= (/) -> k e. ( Evenfin u. Oddfin )) -> ((k +o 1c) =/= (/) -> (k +o 1c) e. ( Evenfin u. Oddfin )))
60	59	a1i 10	. . . . . 6 |- (k e. Nn -> ((k =/= (/) -> k e. ( Evenfin u. Oddfin )) -> ((k +o 1c) =/= (/) -> (k +o 1c) e. ( Evenfin u. Oddfin ))))
61	29, 32, 35, 38, 41, 45, 60	finds 4412	. . . . 5 |- (n e. Nn -> (n =/= (/) -> n e. ( Evenfin u. Oddfin )))
62	61	imp 418	. . . 4 |- ((n e. Nn /\ n =/= (/)) -> n e. ( Evenfin u. Oddfin ))
63	13, 62	sylbi 187	. . 3 |- (n e. ( Nn \ {(/)}) -> n e. ( Evenfin u. Oddfin ))
64	63	ssriv 3278	. 2 |- ( Nn \ {(/)}) C_ ( Evenfin u. Oddfin )
65	12, 64	eqssi 3289	1 |- ( Evenfin u. Oddfin ) = ( Nn \ {(/)})

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  _Vcvv 2860   \ cdif 3207   u. cun 3208   C_ wss 3258  (/)c0 3551  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   Even_fin cevenfin 4437   Odd_fin coddfin 4438

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-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-0c 4378  df-addc 4379  df-nnc 4380  df-evenfin 4445  df-oddfin 4446

This theorem is referenced by:  vinf  4556