Theorem mnfltxr 10014

Description: Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Assertion

Ref	Expression
mnfltxr	⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Proof of Theorem mnfltxr

Step	Hyp	Ref	Expression
1		mnflt 10011	. 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		mnfltpnf 10013	. . 3 ⊢ -∞ < +∞
3		breq2 4090	. . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
4	2, 3	mpbiri 168	. 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴)
5	1, 4	jaoi 721	1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Syntax hints:   → wi 4   ∨ wo 713   = wceq 1395   ∈ wcel 2200   class class class wbr 4086  ℝcr 8024  +∞cpnf 8204  -∞cmnf 8205   < clt 8207

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-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-cnex 8116

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-v 2802  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-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212

This theorem is referenced by:  xrltso  10024