elxrge0 - Intuitionistic Logic Explorer

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Theorem elxrge0 9914

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 970	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞) ↔ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞))
2		0xr 7945	. . 3 ⊢ 0 ∈ ℝ^*
3		pnfxr 7951	. . 3 ⊢ +∞ ∈ ℝ^*
4		elicc1 9860	. . 3 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞)))
5	2, 3, 4	mp2an 423	. 2 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞))
6		pnfge 9725	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7	6	adantr 274	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≤ +∞)
8	7	pm4.71i 389	. 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 968 ∈ wcel 2136 class class class wbr 3982 (class class class)co 5842 0cc0 7753 +∞cpnf 7930 ℝ^*cxr 7932 ≤ cle 7934 [,]cicc 9827

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1re 7847 ax-addrcl 7850 ax-rnegex 7862

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-icc 9831

This theorem is referenced by: 0e0iccpnf 9916 ge0xaddcl 9919 psmetxrge0 12972 isxmet2d 12988 comet 13139 bdxmet 13141

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