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Theorem mnfltpnf 10117

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 2232	. . . 4 ⊢ -∞ = -∞
2		eqid 2232	. . . 4 ⊢ +∞ = +∞
3		olc 719	. . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)))
4	1, 2, 3	mp2an 426	. . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))
5	4	orci 739	. 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <_ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))
6		mnfxr 8329	. . 3 ⊢ -∞ ∈ ℝ^*
7		pnfxr 8325	. . 3 ⊢ +∞ ∈ ℝ^*
8		ltxr 10107	. . 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 716 = wceq 1398 ∈ wcel 2203 class class class wbr 4108 ℝcr 8125 <_ℝ cltrr 8130 +∞cpnf 8304 -∞cmnf 8305 ℝ^*cxr 8306 < clt 8307

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-cnex 8217

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2814 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-br 4109 df-opab 4171 df-xp 4754 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312

This theorem is referenced by: mnfltxr 10118 xrlttr 10127 xrltso 10128 xrlttri3 10129 nltpnft 10146 npnflt 10147 ngtmnft 10149 nmnfgt 10150 xltnegi 10167 xposdif 10214

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