Theorem squeeze0 9074

Description: If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Assertion

Ref	Expression
squeeze0	|- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Distinct variable group:   x, A

Proof of Theorem squeeze0

Step	Hyp	Ref	Expression
1		ltnr 8246	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	3ad2ant1 1042	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. A < A )
3		breq2 4090	. . . . . . 7 |- ( x = A -> ( 0 < x <-> 0 < A ) )
4		breq2 4090	. . . . . . 7 |- ( x = A -> ( A < x <-> A < A ) )
5	3, 4	imbi12d 234	. . . . . 6 |- ( x = A -> ( ( 0 < x -> A < x ) <-> ( 0 < A -> A < A ) ) )
6	5	rspcva 2906	. . . . 5 |- ( ( A e. RR /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
7	6	3adant2 1040	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
8	2, 7	mtod 667	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. 0 < A )
9		simp1 1021	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A e. RR )
10		0red 8170	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 e. RR )
11	9, 10	lenltd 8287	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( A <_ 0 <-> -. 0 < A ) )
12	8, 11	mpbird 167	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A <_ 0 )
13		simp2 1022	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 <_ A )
14	9, 10	letri3d 8285	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( A = 0 <-> ( A <_ 0 /\ 0 <_ A ) ) )
15	12, 13, 14	mpbir2and 950	1 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   class class class wbr 4086   RRcr 8021   0cc0 8022   < clt 8204   <_ cle 8205

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-apti 8137

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210

This theorem is referenced by: (None)