Theorem xrsupexmnf 13319

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 4145	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {-∞}))
2		simpr 483	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦))
3		velsn 4646	. . . . . . . . 9 ⊢ (𝑦 ∈ {-∞} ↔ 𝑦 = -∞)
4		nltmnf 13144	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5		breq2 5153	. . . . . . . . . . 11 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
6	5	notbid 317	. . . . . . . . . 10 ⊢ (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
7	4, 6	syl5ibrcom 246	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦))
8	3, 7	biimtrid 241	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
9	8	adantr 479	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
10	2, 9	jaod 857	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦))
11	1, 10	biimtrid 241	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))
12	11	ex 411	. . . 4 ⊢ (𝑥 ∈ ℝ* → ((𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)))
13	12	ralimdv2 3152	. . 3 ⊢ (𝑥 ∈ ℝ* → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦))
14		elun1 4174	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴 ∪ {-∞}))
15	14	anim1i 613	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧))
16	15	reximi2 3068	. . . . 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 3072	. . 3 ⊢ (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
19	13, 18	anim12d1 608	. 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 3070	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 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   ∪ cun 3942  {csn 4630   class class class wbr 5149  -∞cmnf 11278  ℝ^*cxr 11279   < clt 11280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285

This theorem is referenced by:  xrsupss  13323