Theorem uz11 9673

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 9664	. . . . 5 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
2		eleq2 2269	. . . . . 6 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) <-> M e. ( ZZ>= ` N ) ) )
3		eluzel2 9655	. . . . . 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 9664	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
7		eleq2 2269	. . . . . . . . . . 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 9662	. . . . . . . . . 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 9662	. . . . . . . . . 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 9378	. . . . . . 7 |- ( M e. ZZ -> M e. RR )
19		zre 9378	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
20		letri3 8155	. . . . . . 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 5578	. 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 2176   class class class wbr 4045   ` cfv 5272   RRcr 7926   <_ cle 8110   ZZcz 9374   ZZ>=cuz 9650

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-pre-ltirr 8039  ax-pre-apti 8042

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-ov 5949  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-neg 8248  df-z 9375  df-uz 9651

This theorem is referenced by:  fzopth  10185