Theorem uz11 9706

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 9697	. . . . 5 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
2		eleq2 2271	. . . . . 6 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) <-> M e. ( ZZ>= ` N ) ) )
3		eluzel2 9688	. . . . . 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 9697	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
7		eleq2 2271	. . . . . . . . . . 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 9695	. . . . . . . . . 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 9695	. . . . . . . . . 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 9411	. . . . . . 7 |- ( M e. ZZ -> M e. RR )
19		zre 9411	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
20		letri3 8188	. . . . . . 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 5599	. 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 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290   RRcr 7959   <_ cle 8143   ZZcz 9407   ZZ>=cuz 9683

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-pre-ltirr 8072  ax-pre-apti 8075

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-neg 8281  df-z 9408  df-uz 9684

This theorem is referenced by:  fzopth  10218