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Theorem mnfltpnf 9860

Description: Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnfltpnf	⊢ -∞ < +∞

**Proof of Theorem mnfltpnf**
Step	Hyp	Ref	Expression
1		eqid 2196	. . . 4 ⊢ -∞ = -∞
2		eqid 2196	. . . 4 ⊢ +∞ = +∞
3		olc 712	. . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)))
4	1, 2, 3	mp2an 426	. . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))
5	4	orci 732	. 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))
6		mnfxr 8083	. . 3 ⊢ -∞ ∈ ℝ^*
7		pnfxr 8079	. . 3 ⊢ +∞ ∈ ℝ^*
8		ltxr 9850	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))))
9	6, 7, 8	mp2an 426	. 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))
10	5, 9	mpbir 146	1 ⊢ -∞ < +∞

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2167 class class class wbr 4033 ℝcr 7878 <_ℝ cltrr 7883 +∞cpnf 8058 -∞cmnf 8059 ℝ^*cxr 8060 < clt 8061

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-cnex 7970

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-xp 4669 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066

This theorem is referenced by: mnfltxr 9861 xrlttr 9870 xrltso 9871 xrlttri3 9872 nltpnft 9889 npnflt 9890 ngtmnft 9892 nmnfgt 9893 xltnegi 9910 xposdif 9957

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