Theorem xrinfmexpnf 13327

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 4133	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∪ {+∞}) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {+∞}))
2		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))
3		velsn 4622	. . . . . . . . 9 ⊢ (𝑦 ∈ {+∞} ↔ 𝑦 = +∞)
4		pnfnlt 13149	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
5		breq1 5127	. . . . . . . . . . 11 ⊢ (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
6	5	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
7	4, 6	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → (𝑦 = +∞ → ¬ 𝑦 < 𝑥))
8	3, 7	biimtrid 242	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
10	2, 9	jaod 859	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {+∞}) → ¬ 𝑦 < 𝑥))
11	1, 10	biimtrid 242	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥))
12	11	ex 412	. . . 4 ⊢ (𝑥 ∈ ℝ* → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥)))
13	12	ralimdv2 3150	. . 3 ⊢ (𝑥 ∈ ℝ* → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 → ∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥))
14		elun1 4162	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴 ∪ {+∞}))
15	14	anim1i 615	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 < 𝑦) → (𝑧 ∈ (𝐴 ∪ {+∞}) ∧ 𝑧 < 𝑦))
16	15	reximi2 3070	. . . . 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 3074	. . 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 3072	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 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ∪ cun 3929  {csn 4606   class class class wbr 5124  +∞cpnf 11271  ℝ^*cxr 11273   < clt 11274

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-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279

This theorem is referenced by:  xrinfmss  13331