Theorem nn0lt2 9560

Description: A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

Assertion

Ref	Expression
nn0lt2	|- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) )

Proof of Theorem nn0lt2

Step	Hyp	Ref	Expression
1		olc 718	. . 3 |- ( N = 1 -> ( N = 0 \/ N = 1 ) )
2	1	a1i 9	. 2 |- ( ( N e. NN0 /\ N < 2 ) -> ( N = 1 -> ( N = 0 \/ N = 1 ) ) )
3		nn0z 9498	. . . . . 6 |- ( N e. NN0 -> N e. ZZ )
4		2z 9506	. . . . . 6 |- 2 e. ZZ
5		zltlem1 9536	. . . . . 6 |- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> N <_ ( 2 - 1 ) ) )
6	3, 4, 5	sylancl 413	. . . . 5 |- ( N e. NN0 -> ( N < 2 <-> N <_ ( 2 - 1 ) ) )
7		2m1e1 9260	. . . . . 6 |- ( 2 - 1 ) = 1
8	7	breq2i 4096	. . . . 5 |- ( N <_ ( 2 - 1 ) <-> N <_ 1 )
9	6, 8	bitrdi 196	. . . 4 |- ( N e. NN0 -> ( N < 2 <-> N <_ 1 ) )
10		necom 2486	. . . . 5 |- ( N =/= 1 <-> 1 =/= N )
11		1z 9504	. . . . . . . 8 |- 1 e. ZZ
12		zltlen 9557	. . . . . . . 8 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> ( N < 1 <-> ( N <_ 1 /\ 1 =/= N ) ) )
13	3, 11, 12	sylancl 413	. . . . . . 7 |- ( N e. NN0 -> ( N < 1 <-> ( N <_ 1 /\ 1 =/= N ) ) )
14		nn0lt10b 9559	. . . . . . . . . 10 |- ( N e. NN0 -> ( N < 1 <-> N = 0 ) )
15	14	biimpa 296	. . . . . . . . 9 |- ( ( N e. NN0 /\ N < 1 ) -> N = 0 )
16	15	orcd 740	. . . . . . . 8 |- ( ( N e. NN0 /\ N < 1 ) -> ( N = 0 \/ N = 1 ) )
17	16	ex 115	. . . . . . 7 |- ( N e. NN0 -> ( N < 1 -> ( N = 0 \/ N = 1 ) ) )
18	13, 17	sylbird 170	. . . . . 6 |- ( N e. NN0 -> ( ( N <_ 1 /\ 1 =/= N ) -> ( N = 0 \/ N = 1 ) ) )
19	18	expd 258	. . . . 5 |- ( N e. NN0 -> ( N <_ 1 -> ( 1 =/= N -> ( N = 0 \/ N = 1 ) ) ) )
20	10, 19	syl7bi 165	. . . 4 |- ( N e. NN0 -> ( N <_ 1 -> ( N =/= 1 -> ( N = 0 \/ N = 1 ) ) ) )
21	9, 20	sylbid 150	. . 3 |- ( N e. NN0 -> ( N < 2 -> ( N =/= 1 -> ( N = 0 \/ N = 1 ) ) ) )
22	21	imp 124	. 2 |- ( ( N e. NN0 /\ N < 2 ) -> ( N =/= 1 -> ( N = 0 \/ N = 1 ) ) )
23		zdceq 9554	. . . . 5 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> DECID N = 1 )
24	3, 11, 23	sylancl 413	. . . 4 |- ( N e. NN0 -> DECID N = 1 )
25	24	adantr 276	. . 3 |- ( ( N e. NN0 /\ N < 2 ) -> DECID N = 1 )
26		dcne 2413	. . 3 |- (DECID N = 1 <-> ( N = 1 \/ N =/= 1 ) )
27	25, 26	sylib 122	. 2 |- ( ( N e. NN0 /\ N < 2 ) -> ( N = 1 \/ N =/= 1 ) )
28	2, 22, 27	mpjaod 725	1 |- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088  (class class class)co 6017   0cc0 8031   1c1 8032   < clt 8213   <_ cle 8214   - cmin 8349   2c2 9193   NN0cn0 9401   ZZcz 9478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479

This theorem is referenced by: (None)