Theorem xrinfmexpnf 13040

Description: Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)

Assertion

Ref	Expression
xrinfmexpnf	⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrinfmexpnf

Step	Hyp	Ref	Expression
1		elun 4083	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∪ {+∞}) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {+∞}))
2		simpr 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))
3		velsn 4577	. . . . . . . . 9 ⊢ (𝑦 ∈ {+∞} ↔ 𝑦 = +∞)
4		pnfnlt 12864	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
5		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
6	5	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
7	4, 6	syl5ibrcom 246	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → (𝑦 = +∞ → ¬ 𝑦 < 𝑥))
8	3, 7	syl5bi 241	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
9	8	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
10	2, 9	jaod 856	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {+∞}) → ¬ 𝑦 < 𝑥))
11	1, 10	syl5bi 241	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥))
12	11	ex 413	. . . 4 ⊢ (𝑥 ∈ ℝ* → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥)))
13	12	ralimdv2 3107	. . 3 ⊢ (𝑥 ∈ ℝ* → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 → ∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥))
14		elun1 4110	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴 ∪ {+∞}))
15	14	anim1i 615	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 < 𝑦) → (𝑧 ∈ (𝐴 ∪ {+∞}) ∧ 𝑧 < 𝑦))
16	15	reximi2 3175	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)
17	16	imim2i 16	. . . 4 ⊢ ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
18	17	ralimi 3087	. . 3 ⊢ (∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
19	13, 18	anim12d1 610	. 2 ⊢ (𝑥 ∈ ℝ* → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))))
20	19	reximia 3176	1 ⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∪ cun 3885  {csn 4561   class class class wbr 5074  +∞cpnf 11006  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-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  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-xp 5595  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014

This theorem is referenced by:  xrinfmss  13044