supxrmnf2 - Mathbox for Glauco Siliprandi

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Theorem supxrmnf2 42973

Description: Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Assertion**
Ref	Expression
supxrmnf2	⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

**Proof of Theorem supxrmnf2**
Step	Hyp	Ref	Expression
1		ssdifss 4070	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
2		supxrmnf 13051	. . . . 5 ⊢ ((𝐴 ∖ {-∞}) ⊆ ℝ^* → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
3	1, 2	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
4	3	adantr 481	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
5		difsnid 4743	. . . . 5 ⊢ (-∞ ∈ 𝐴 → ((𝐴 ∖ {-∞}) ∪ {-∞}) = 𝐴)
6	5	supeq1d 9205	. . . 4 ⊢ (-∞ ∈ 𝐴 → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
7	6	adantl 482	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
8	4, 7	eqtr3d 2780	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
9		difsn 4731	. . . 4 ⊢ (¬ -∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) = 𝐴)
10	9	supeq1d 9205	. . 3 ⊢ (¬ -∞ ∈ 𝐴 → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
11	10	adantl 482	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ ¬ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
12	8, 11	pm2.61dan 810	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∖ cdif 3884 ∪ cun 3885 ⊆ wss 3887 {csn 4561 supcsup 9199 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009

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: supminfxr2 43009

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