infxrpnf2 - Mathbox for Glauco Siliprandi

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Theorem infxrpnf2 45446

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

**Assertion**
Ref	Expression
infxrpnf2	⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

**Proof of Theorem infxrpnf2**
Step	Hyp	Ref	Expression
1		ssdifss 4093	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {+∞}) ⊆ ℝ^*)
2		infxrpnf 45429	. . . . 5 ⊢ ((𝐴 ∖ {+∞}) ⊆ ℝ^* → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
3	1, 2	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
5		difsnid 4764	. . . . 5 ⊢ (+∞ ∈ 𝐴 → ((𝐴 ∖ {+∞}) ∪ {+∞}) = 𝐴)
6	5	infeq1d 9387	. . . 4 ⊢ (+∞ ∈ 𝐴 → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
7	6	adantl 481	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
8	4, 7	eqtr3d 2766	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
9		difsn 4752	. . . 4 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {+∞}) = 𝐴)
10	9	infeq1d 9387	. . 3 ⊢ (¬ +∞ ∈ 𝐴 → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
11	10	adantl 481	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ ¬ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
12	8, 11	pm2.61dan 812	1 ⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3902 ∪ cun 3903 ⊆ wss 3905 {csn 4579 infcinf 9350 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368

This theorem is referenced by: supminfxr2 45452

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