Theorem xrsupexmnf 12686

Description: Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrsupexmnf

Step	Hyp	Ref	Expression
1		elun 4076	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {-∞}))
2		simpr 488	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦))
3		velsn 4541	. . . . . . . . 9 ⊢ (𝑦 ∈ {-∞} ↔ 𝑦 = -∞)
4		nltmnf 12512	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5		breq2 5034	. . . . . . . . . . 11 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
6	5	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
7	4, 6	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦))
8	3, 7	syl5bi 245	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
9	8	adantr 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
10	2, 9	jaod 856	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦))
11	1, 10	syl5bi 245	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))
12	11	ex 416	. . . 4 ⊢ (𝑥 ∈ ℝ* → ((𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)))
13	12	ralimdv2 3143	. . 3 ⊢ (𝑥 ∈ ℝ* → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦))
14		elun1 4103	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴 ∪ {-∞}))
15	14	anim1i 617	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧))
16	15	reximi2 3207	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)
17	16	imim2i 16	. . . 4 ⊢ ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
18	17	ralimi 3128	. . 3 ⊢ (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
19	13, 18	anim12d1 612	. 2 ⊢ (𝑥 ∈ ℝ* → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))))
20	19	reximia 3205	1 ⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∪ cun 3879  {csn 4525   class class class wbr 5030  -∞cmnf 10662  ℝ^*cxr 10663   < clt 10664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669

This theorem is referenced by:  xrsupss  12690