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Theorem xrsupssd 13349

Description: Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)

**Hypotheses**
Ref	Expression
xrsupssd.1	⊢ (𝜑 → 𝐵 ⊆ 𝐶)
xrsupssd.2	⊢ (𝜑 → 𝐶 ⊆ ℝ^*)

**Assertion**
Ref	Expression
xrsupssd	⊢ (𝜑 → sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ))

Proof of Theorem xrsupssd Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 13157	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		xrsupssd.1	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4		xrsupssd.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ ℝ^*)
5	3, 4	sstrd 3969	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
6		xrsupss 13325	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))
8		xrsupss 13325	. . . 4 ⊢ (𝐶 ⊆ ℝ^* → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐶 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐶 𝑦 < 𝑧)))
9	4, 8	syl 17	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐶 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ^* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐶 𝑦 < 𝑧)))
10	2, 3, 4, 7, 9	supssd 9475	. 2 ⊢ (𝜑 → ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < ))
11	2, 7	supcl 9470	. . 3 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
12	2, 9	supcl 9470	. . 3 ⊢ (𝜑 → sup(𝐶, ℝ^, < ) ∈ ℝ^)
13		xrlenlt 11300	. . 3 ⊢ ((sup(𝐵, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐶, ℝ^, < ) ∈ ℝ^) → (sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ) ↔ ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < )))
14	11, 12, 13	syl2anc 584	. 2 ⊢ (𝜑 → (sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ) ↔ ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < )))
15	10, 14	mpbird 257	1 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 ⊆ wss 3926 class class class wbr 5119 Or wor 5560 supcsup 9452 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-sup 9454 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469

This theorem is referenced by: liminflelimsuplem 45804

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