Theorem icossicc 9584

Description: A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)

Assertion

Ref	Expression
icossicc	⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)

Proof of Theorem icossicc

Dummy variables 𝑎 𝑏 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ico 9518	. 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)})
2		df-icc 9519	. 2 ⊢ [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 ≤ 𝑏)})
3		idd 21	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 ≤ 𝑤 → 𝐴 ≤ 𝑤))
4		xrltle 9425	. 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
5	1, 2, 3, 4	ixxssixx 9526	1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)

Syntax hints:   ∧ wa 103   ∈ wcel 1448   ⊆ wss 3021   class class class wbr 3875  (class class class)co 5706  ℝ^*cxr 7671   < clt 7672   ≤ cle 7673  [,)cico 9514  [,]cicc 9515

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-ltirr 7607  ax-pre-lttrn 7609

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-ico 9518  df-icc 9519

This theorem is referenced by: (None)