elxrge0 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > elxrge0		GIF version

Theorem elxrge0 10162

Description: Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

**Assertion**
Ref	Expression
elxrge0	⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴))

**Proof of Theorem elxrge0**
Step	Hyp	Ref	Expression
1		df-3an 1004	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
2		0xr 8181	. . 3 ⊢ 0 ∈ ℝ^*
3		pnfxr 8187	. . 3 ⊢ +∞ ∈ ℝ^*
4		elicc1 10108	. . 3 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞)))
5	2, 3, 4	mp2an 426	. 2 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞))
6		pnfge 9973	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7	6	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≤ +∞)
8	7	pm4.71i 391	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
9	1, 5, 8	3bitr4i 212	1 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 1002 ∈ wcel 2200 class class class wbr 4082 (class class class)co 5994 0cc0 7987 +∞cpnf 8166 ℝ^*cxr 8168 ≤ cle 8170 [,]cicc 10075

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 617 ax-in2 618 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 4201 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-cnex 8078 ax-resscn 8079 ax-1re 8081 ax-addrcl 8084 ax-rnegex 8096

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 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-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-opab 4145 df-id 4381 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-iota 5274 df-fun 5316 df-fv 5322 df-ov 5997 df-oprab 5998 df-mpo 5999 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-icc 10079

This theorem is referenced by: 0e0iccpnf 10164 ge0xaddcl 10167 psmetxrge0 14991 isxmet2d 15007 comet 15158 bdxmet 15160

W3C validator