xrsupssd - Mathbox for Thierry Arnoux

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Theorem xrsupssd 31972

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 13120	. . . 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 3993	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
6		xrsupss 13288	. . . 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 13288	. . . 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 31934	. 2 ⊢ (𝜑 → ¬ sup(𝐶, ℝ^, < ) < sup(𝐵, ℝ^, < ))
11	2, 7	supcl 9453	. . 3 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
12	2, 9	supcl 9453	. . 3 ⊢ (𝜑 → sup(𝐶, ℝ^, < ) ∈ ℝ^)
13		xrlenlt 11279	. . 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 585	. 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 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 Or wor 5588 supcsup 9435 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447

This theorem is referenced by: (None)

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