supxrmnf2 - Mathbox for Glauco Siliprandi

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Theorem supxrmnf2 45402

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 4099	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {-∞}) ⊆ ℝ^*)
2		supxrmnf 13253	. . . . 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 480	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup((𝐴 ∖ {-∞}), ℝ^, < ))
5		difsnid 4770	. . . . 5 ⊢ (-∞ ∈ 𝐴 → ((𝐴 ∖ {-∞}) ∪ {-∞}) = 𝐴)
6	5	supeq1d 9373	. . . 4 ⊢ (-∞ ∈ 𝐴 → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
7	6	adantl 481	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
8	4, 7	eqtr3d 2766	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
9		difsn 4758	. . . 4 ⊢ (¬ -∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) = 𝐴)
10	9	supeq1d 9373	. . 3 ⊢ (¬ -∞ ∈ 𝐴 → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
11	10	adantl 481	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ ¬ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))
12	8, 11	pm2.61dan 812	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∖ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 {csn 4585 supcsup 9367 -∞cmnf 11182 ℝ^*cxr 11183 < clt 11184

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384

This theorem is referenced by: supminfxr2 45438

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