Theorem uz11 9877

Description: The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)

Assertion

Ref	Expression
uz11	|- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )

Proof of Theorem uz11

Step	Hyp	Ref	Expression
1		uzid 9868	. . . . 5 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
2		eleq2 2296	. . . . . 6 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) <-> M e. ( ZZ>= ` N ) ) )
3		eluzel2 9858	. . . . . 6 |- ( M e. ( ZZ>= ` N ) -> N e. ZZ )
4	2, 3	biimtrdi 163	. . . . 5 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) -> N e. ZZ ) )
5	1, 4	mpan9 281	. . . 4 |- ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) -> N e. ZZ )
6		uzid 9868	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
7		eleq2 2296	. . . . . . . . . . 11 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ( ZZ>= ` M ) <-> N e. ( ZZ>= ` N ) ) )
8	6, 7	imbitrrid 156	. . . . . . . . . 10 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ZZ -> N e. ( ZZ>= ` M ) ) )
9		eluzle 9866	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
10	8, 9	syl6 33	. . . . . . . . 9 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ZZ -> M <_ N ) )
11	1, 2	imbitrid 154	. . . . . . . . . 10 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ZZ -> M e. ( ZZ>= ` N ) ) )
12		eluzle 9866	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
13	11, 12	syl6 33	. . . . . . . . 9 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ZZ -> N <_ M ) )
14	10, 13	anim12d 335	. . . . . . . 8 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( ( N e. ZZ /\ M e. ZZ ) -> ( M <_ N /\ N <_ M ) ) )
15	14	impl 380	. . . . . . 7 |- ( ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ N e. ZZ ) /\ M e. ZZ ) -> ( M <_ N /\ N <_ M ) )
16	15	ancoms 268	. . . . . 6 |- ( ( M e. ZZ /\ ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ N e. ZZ ) ) -> ( M <_ N /\ N <_ M ) )
17	16	anassrs 400	. . . . 5 |- ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> ( M <_ N /\ N <_ M ) )
18		zre 9581	. . . . . . 7 |- ( M e. ZZ -> M e. RR )
19		zre 9581	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
20		letri3 8354	. . . . . . 7 |- ( ( M e. RR /\ N e. RR ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
21	18, 19, 20	syl2an 289	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
22	21	adantlr 477	. . . . 5 |- ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
23	17, 22	mpbird 167	. . . 4 |- ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> M = N )
24	5, 23	mpdan 421	. . 3 |- ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) -> M = N )
25	24	ex 115	. 2 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> M = N ) )
26		fveq2 5670	. 2 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
27	25, 26	impbid1 142	1 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352   RRcr 8126   <_ cle 8309   ZZcz 9577   ZZ>=cuz 9853

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239  ax-pre-apti 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-neg 8447  df-z 9578  df-uz 9854

This theorem is referenced by:  fzopth  10395