elxrge0 - Intuitionistic Logic Explorer

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Theorem elxrge0 9754

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 964	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
2		0xr 7805	. . 3 ⊢ 0 ∈ ℝ^*
3		pnfxr 7811	. . 3 ⊢ +∞ ∈ ℝ^*
4		elicc1 9700	. . 3 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞)))
5	2, 3, 4	mp2an 422	. 2 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞))
6		pnfge 9568	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7	6	adantr 274	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≤ +∞)
8	7	pm4.71i 388	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
9	1, 5, 8	3bitr4i 211	1 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴))

Colors of variables: wff set class

Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 0cc0 7613 +∞cpnf 7790 ℝ^*cxr 7792 ≤ cle 7794 [,]cicc 9667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-icc 9671

This theorem is referenced by: 0e0iccpnf 9756 ge0xaddcl 9759 psmetxrge0 12490 isxmet2d 12506 comet 12657 bdxmet 12659

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