Theorem uz11 9624

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 9615	. . . . 5 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
2		eleq2 2260	. . . . . 6 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) <-> M e. ( ZZ>= ` N ) ) )
3		eluzel2 9606	. . . . . 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 9615	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
7		eleq2 2260	. . . . . . . . . . 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 9613	. . . . . . . . . 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 9613	. . . . . . . . . 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 9330	. . . . . . 7 |- ( M e. ZZ -> M e. RR )
19		zre 9330	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
20		letri3 8107	. . . . . . 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 5558	. 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 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258   RRcr 7878   <_ cle 8062   ZZcz 9326   ZZ>=cuz 9601

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991  ax-pre-apti 7994

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-neg 8200  df-z 9327  df-uz 9602

This theorem is referenced by:  fzopth  10136