supxrmnf2 - Mathbox for Glauco Siliprandi

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Theorem supxrmnf2 42944

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 4075	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
2		supxrmnf 13050	. . . . 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 4749	. . . . 5 ⊢ (-∞ ∈ 𝐴 → ((𝐴 ∖ {-∞}) ∪ {-∞}) = 𝐴)
6	5	supeq1d 9183	. . . 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 2782	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
9		difsn 4737	. . . 4 ⊢ (¬ -∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) = 𝐴)
10	9	supeq1d 9183	. . 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 1542 ∈ wcel 2110 ∖ cdif 3889 ∪ cun 3890 ⊆ wss 3892 {csn 4567 supcsup 9177 -∞cmnf 11008 ℝ^*cxr 11009 < clt 11010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208

This theorem is referenced by: supminfxr2 42980

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