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Theorem supxrss 12887

Description: Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)

**Assertion**
Ref	Expression
supxrss	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))

Proof of Theorem supxrss Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ℝ^)
2		simpl 486	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) → 𝐴 ⊆ 𝐵)
3	2	sselda 3887	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
4		supxrub 12879	. . . 4 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ sup(𝐵, ℝ^*, < ))
5	1, 3, 4	syl2anc 587	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐵, ℝ^, < ))
6	5	ralrimiva 3095	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ^, < ))
7		sstr 3895	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
8		supxrcl 12870	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
9	8	adantl 485	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → sup(𝐵, ℝ^, < ) ∈ ℝ^*)
10		supxrleub 12881	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐵, ℝ^, < ) ∈ ℝ^) → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ^*, < )))
11	7, 9, 10	syl2anc 587	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ^, < )))
12	6, 11	mpbird 260	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 class class class wbr 5039 supcsup 9034 ℝ^*cxr 10831 < clt 10832 ≤ cle 10833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030

This theorem is referenced by: deg1mul3le 24968 ioossioobi 42671 limsupres 42864 supcnvlimsup 42899 liminfval2 42927 liminflelimsuplem 42934 sge0less 43548 sge0reuz 43603 smflimsuplem4 43971

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