Theorem nmnfgt 9942

Description: An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)

Assertion

Ref	Expression
nmnfgt	|- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )

Proof of Theorem nmnfgt

Step	Hyp	Ref	Expression
1		ngtmnft 9941	. . . 4 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
2	1	biimpd 144	. . 3 |- ( A e. RR* -> ( A = -oo -> -. -oo < A ) )
3	2	necon2ad 2433	. 2 |- ( A e. RR* -> ( -oo < A -> A =/= -oo ) )
4		mnflt 9907	. . . . 5 |- ( A e. RR -> -oo < A )
5	4	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A e. RR ) -> -oo < A )
6		mnfltpnf 9909	. . . . . 6 |- -oo < +oo
7		breq2 4049	. . . . . 6 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
8	6, 7	mpbiri 168	. . . . 5 |- ( A = +oo -> -oo < A )
9	8	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = +oo ) -> -oo < A )
10		simpr 110	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A = -oo )
11		simplr 528	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A =/= -oo )
12	10, 11	pm2.21ddne 2459	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> -oo < A )
13		elxr 9900	. . . . . 6 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
14	13	biimpi 120	. . . . 5 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
15	14	adantr 276	. . . 4 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
16	5, 9, 12, 15	mpjao3dan 1320	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) -> -oo < A )
17	16	ex 115	. 2 |- ( A e. RR* -> ( A =/= -oo -> -oo < A ) )
18	3, 17	impbid 129	1 |- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 980   = wceq 1373   e. wcel 2176   =/= wne 2376   class class class wbr 4045   RRcr 7926   +oocpnf 8106   -oocmnf 8107   RR*cxr 8108   < clt 8109

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

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-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-xp 4682  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114

This theorem is referenced by:  xlt2add  10004  xrmaxadd  11605  xblpnfps  14903  xblpnf  14904