Theorem xrltne 10047

Description: 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

Assertion

Ref	Expression
xrltne	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A )

Proof of Theorem xrltne

Step	Hyp	Ref	Expression
1		xrltnr 10013	. . . . 5 |- ( A e. RR* -> -. A < A )
2		breq2 4092	. . . . . 6 |- ( B = A -> ( A < B <-> A < A ) )
3	2	notbid 673	. . . . 5 |- ( B = A -> ( -. A < B <-> -. A < A ) )
4	1, 3	syl5ibrcom 157	. . . 4 |- ( A e. RR* -> ( B = A -> -. A < B ) )
5	4	necon2ad 2459	. . 3 |- ( A e. RR* -> ( A < B -> B =/= A ) )
6	5	imp 124	. 2 |- ( ( A e. RR* /\ A < B ) -> B =/= A )
7	6	3adant2 1042	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088   RR*cxr 8212   < clt 8213

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218

This theorem is referenced by: (None)