xrsupssd - Mathbox for Thierry Arnoux

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Theorem xrsupssd 31082

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 12875	. . . 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 3931	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
6		xrsupss 13043	. . . 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 13043	. . . 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 31044	. 2 ⊢ (𝜑 → ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < ))
11	2, 7	supcl 9217	. . 3 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
12	2, 9	supcl 9217	. . 3 ⊢ (𝜑 → sup(𝐶, ℝ^, < ) ∈ ℝ^)
13		xrlenlt 11040	. . 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 256	1 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ≤ sup(𝐶, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 Or wor 5502 supcsup 9199 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208

This theorem is referenced by: (None)

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