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Theorem supxrss 13489

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 781	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ℝ^)
2		simpl 488	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) → 𝐴 ⊆ 𝐵)
3	2	sselda 3913	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
4		supxrub 13481	. . . 4 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ sup(𝐵, ℝ^*, < ))
5	1, 3, 4	syl2anc 596	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐵, ℝ^, < ))
6	5	ralrimiva 3158	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ^, < ))
7		sstr 3921	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
8		supxrcl 13472	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
9	8	adantl 487	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → sup(𝐵, ℝ^, < ) ∈ ℝ^*)
10		supxrleub 13483	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐵, ℝ^, < ) ∈ ℝ^) → (sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ^*, < )))
11	7, 9, 10	syl2anc 596	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∈ wcel 2147 ∀wral 3080 ⊆ wss 3900 class class class wbr 5106 supcsup 9459 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-sup 9461 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571

This theorem is used by: deg1mul3le 26465 ioossioobi 46734 limsupres 46920 supcnvlimsup 46955 liminfval2 46983 sge0less 47607 sge0reuz 47662 smflimsuplem4 48038

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