Proof of Theorem xrinfmsslem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | raleq 3323 | . . . . . 6
⊢ (𝐴 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥)) | 
| 2 |  | rexeq 3322 | . . . . . . . 8
⊢ (𝐴 = ∅ → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) | 
| 3 | 2 | imbi2d 340 | . . . . . . 7
⊢ (𝐴 = ∅ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) | 
| 4 | 3 | ralbidv 3178 | . . . . . 6
⊢ (𝐴 = ∅ → (∀𝑦 ∈ ℝ*
(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) | 
| 5 | 1, 4 | anbi12d 632 | . . . . 5
⊢ (𝐴 = ∅ →
((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))) | 
| 6 | 5 | rexbidv 3179 | . . . 4
⊢ (𝐴 = ∅ → (∃𝑥 ∈ ℝ*
(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))) | 
| 7 |  | infm3 12227 | . . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 8 |  | rexr 11307 | . . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) | 
| 9 | 8 | anim1i 615 | . . . . . . . . 9
⊢ ((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)))) | 
| 10 | 9 | reximi2 3079 | . . . . . . . 8
⊢
(∃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))) | 
| 11 | 7, 10 | syl 17 | . . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 12 |  | elxr 13158 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ*
↔ (𝑦 ∈ ℝ
∨ 𝑦 = +∞ ∨
𝑦 =
-∞)) | 
| 13 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 14 |  | ssel 3977 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ)) | 
| 15 |  | ltpnf 13162 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ ℝ → 𝑧 < +∞) | 
| 16 | 14, 15 | syl6 35 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 < +∞)) | 
| 17 | 16 | ancld 550 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 < +∞))) | 
| 18 | 17 | eximdv 1917 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ⊆ ℝ →
(∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < +∞))) | 
| 19 |  | n0 4353 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝐴) | 
| 20 |  | df-rex 3071 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(∃𝑧 ∈
𝐴 𝑧 < +∞ ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < +∞)) | 
| 21 | 18, 19, 20 | 3imtr4g 296 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ →
∃𝑧 ∈ 𝐴 𝑧 < +∞)) | 
| 22 | 21 | imp 406 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) →
∃𝑧 ∈ 𝐴 𝑧 < +∞) | 
| 23 | 22 | a1d 25 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (𝑥 < +∞ →
∃𝑧 ∈ 𝐴 𝑧 < +∞)) | 
| 24 | 23 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ 𝑦 = +∞) →
(𝑥 < +∞ →
∃𝑧 ∈ 𝐴 𝑧 < +∞)) | 
| 25 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = +∞ → (𝑥 < 𝑦 ↔ 𝑥 < +∞)) | 
| 26 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = +∞ → (𝑧 < 𝑦 ↔ 𝑧 < +∞)) | 
| 27 | 26 | rexbidv 3179 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = +∞ → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑧 < +∞)) | 
| 28 | 25, 27 | imbi12d 344 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = +∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))) | 
| 29 | 28 | adantl 481 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ 𝑦 = +∞) →
((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))) | 
| 30 | 24, 29 | mpbird 257 | . . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ 𝑦 = +∞) →
(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 31 | 30 | ex 412 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ (𝑦 = +∞ →
(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 32 | 31 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 = +∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 33 |  | nltmnf 13171 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℝ*
→ ¬ 𝑥 <
-∞) | 
| 34 | 33 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = -∞) →
¬ 𝑥 <
-∞) | 
| 35 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞)) | 
| 36 | 35 | notbid 318 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞)) | 
| 37 | 36 | adantl 481 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = -∞) →
(¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞)) | 
| 38 | 34, 37 | mpbird 257 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = -∞) →
¬ 𝑥 < 𝑦) | 
| 39 | 38 | pm2.21d 121 | . . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = -∞) →
(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 40 | 39 | ex 412 | . . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ*
→ (𝑦 = -∞ →
(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 41 | 40 | ad2antlr 727 | . . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 = -∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 42 | 13, 32, 41 | 3jaod 1431 | . . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 43 | 12, 42 | biimtrid 242 | . . . . . . . . . . . 12
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 ∈ ℝ* → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 44 | 43 | ex 412 | . . . . . . . . . . 11
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ ((𝑦 ∈ ℝ
→ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → (𝑦 ∈ ℝ* → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 45 | 44 | ralimdv2 3163 | . . . . . . . . . 10
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ (∀𝑦 ∈
ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 46 | 45 | anim2d 612 | . . . . . . . . 9
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 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)))) | 
| 47 | 46 | reximdva 3168 | . . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) →
(∃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)))) | 
| 48 | 47 | 3adant3 1133 | . . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (∃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)))) | 
| 49 | 11, 48 | mpd 15 | . . . . . 6
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 50 | 49 | 3expa 1119 | . . . . 5
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 51 |  | ralnex 3072 | . . . . . . . . 9
⊢
(∀𝑥 ∈
ℝ ¬ ∀𝑦
∈ 𝐴 𝑥 ≤ 𝑦 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | 
| 52 |  | rexnal 3100 | . . . . . . . . . . . 12
⊢
(∃𝑦 ∈
𝐴 ¬ 𝑥 ≤ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | 
| 53 |  | ssel2 3978 | . . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ) | 
| 54 |  | letric 11361 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) | 
| 55 | 54 | ancoms 458 | . . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) | 
| 56 | 55 | ord 865 | . . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (¬
𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥)) | 
| 57 | 53, 56 | sylan 580 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥)) | 
| 58 | 57 | an32s 652 | . . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥)) | 
| 59 | 58 | reximdva 3168 | . . . . . . . . . . . 12
⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) →
(∃𝑦 ∈ 𝐴 ¬ 𝑥 ≤ 𝑦 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) | 
| 60 | 52, 59 | biimtrrid 243 | . . . . . . . . . . 11
⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (¬
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) | 
| 61 | 60 | ralimdva 3167 | . . . . . . . . . 10
⊢ (𝐴 ⊆ ℝ →
(∀𝑥 ∈ ℝ
¬ ∀𝑦 ∈
𝐴 𝑥 ≤ 𝑦 → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) | 
| 62 | 61 | imp 406 | . . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
¬ ∀𝑦 ∈
𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | 
| 63 | 51, 62 | sylan2br 595 | . . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | 
| 64 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑥 ↔ 𝑧 ≤ 𝑥)) | 
| 65 | 64 | cbvrexvw 3238 | . . . . . . . . 9
⊢
(∃𝑦 ∈
𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) | 
| 66 | 65 | ralbii 3093 | . . . . . . . 8
⊢
(∀𝑥 ∈
ℝ ∃𝑦 ∈
𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) | 
| 67 | 63, 66 | sylib 218 | . . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) | 
| 68 |  | mnfxr 11318 | . . . . . . . 8
⊢ -∞
∈ ℝ* | 
| 69 |  | ssel 3977 | . . . . . . . . . . . 12
⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) | 
| 70 |  | rexr 11307 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℝ*) | 
| 71 |  | nltmnf 13171 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ*
→ ¬ 𝑦 <
-∞) | 
| 72 | 70, 71 | syl 17 | . . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ → ¬
𝑦 <
-∞) | 
| 73 | 69, 72 | syl6 35 | . . . . . . . . . . 11
⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < -∞)) | 
| 74 | 73 | ralrimiv 3145 | . . . . . . . . . 10
⊢ (𝐴 ⊆ ℝ →
∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞) | 
| 75 | 74 | adantr 480 | . . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞) | 
| 76 |  | peano2rem 11576 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈
ℝ) | 
| 77 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = (𝑦 − 1) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (𝑦 − 1))) | 
| 78 | 77 | rexbidv 3179 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (𝑦 − 1) → (∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))) | 
| 79 | 78 | rspcva 3620 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑦 − 1) ∈ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1)) | 
| 80 | 79 | adantrr 717 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 − 1) ∈ ℝ ∧
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ)) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1)) | 
| 81 | 80 | ancoms 458 | . . . . . . . . . . . . . . . . . . 19
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ (𝑦 − 1) ∈ ℝ) →
∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1)) | 
| 82 | 76, 81 | sylan2 593 | . . . . . . . . . . . . . . . . . 18
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1)) | 
| 83 |  | ssel2 3978 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ) | 
| 84 |  | ltm1 12109 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ℝ → (𝑦 − 1) < 𝑦) | 
| 85 | 84 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) < 𝑦) | 
| 86 | 76 | ancri 549 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ℝ → ((𝑦 − 1) ∈ ℝ ∧
𝑦 ∈
ℝ)) | 
| 87 |  | lelttr 11351 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑧 ∈ ℝ ∧ (𝑦 − 1) ∈ ℝ ∧
𝑦 ∈ ℝ) →
((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦)) | 
| 88 | 87 | 3expb 1121 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑧 ∈ ℝ ∧ ((𝑦 − 1) ∈ ℝ ∧
𝑦 ∈ ℝ)) →
((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦)) | 
| 89 | 86, 88 | sylan2 593 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦)) | 
| 90 | 85, 89 | mpan2d 694 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦)) | 
| 91 | 83, 90 | sylan 580 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦)) | 
| 92 | 91 | an32s 652 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦)) | 
| 93 | 92 | reximdva 3168 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) →
(∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 94 | 93 | adantll 714 | . . . . . . . . . . . . . . . . . 18
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 95 | 82, 94 | mpd 15 | . . . . . . . . . . . . . . . . 17
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) | 
| 96 | 95 | exp31 419 | . . . . . . . . . . . . . . . 16
⊢
(∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 97 | 96 | a1dd 50 | . . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 98 | 97 | com4r 94 | . . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 99 |  | 0re 11263 | . . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
ℝ | 
| 100 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 0 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 0)) | 
| 101 | 100 | rexbidv 3179 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 0 → (∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ 0)) | 
| 102 | 101 | rspcva 3620 | . . . . . . . . . . . . . . . . . . 19
⊢ ((0
∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 𝑧 ≤ 0) | 
| 103 | 99, 102 | mpan 690 | . . . . . . . . . . . . . . . . . 18
⊢
(∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 → ∃𝑧 ∈ 𝐴 𝑧 ≤ 0) | 
| 104 | 83, 15 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 < +∞) | 
| 105 | 104 | a1d 25 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ 0 → 𝑧 < +∞)) | 
| 106 | 105 | reximdva 3168 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ⊆ ℝ →
(∃𝑧 ∈ 𝐴 𝑧 ≤ 0 → ∃𝑧 ∈ 𝐴 𝑧 < +∞)) | 
| 107 | 103, 106 | mpan9 506 | . . . . . . . . . . . . . . . . 17
⊢
((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < +∞) | 
| 108 | 107, 27 | imbitrrid 246 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = +∞ →
((∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 109 | 108 | a1dd 50 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = +∞ →
((∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → (-∞ <
𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 110 | 109 | expd 415 | . . . . . . . . . . . . . 14
⊢ (𝑦 = +∞ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 111 |  | xrltnr 13161 | . . . . . . . . . . . . . . . . . 18
⊢ (-∞
∈ ℝ* → ¬ -∞ <
-∞) | 
| 112 | 68, 111 | ax-mp 5 | . . . . . . . . . . . . . . . . 17
⊢  ¬
-∞ < -∞ | 
| 113 |  | breq2 5147 | . . . . . . . . . . . . . . . . 17
⊢ (𝑦 = -∞ → (-∞
< 𝑦 ↔ -∞ <
-∞)) | 
| 114 | 112, 113 | mtbiri 327 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = -∞ → ¬
-∞ < 𝑦) | 
| 115 | 114 | pm2.21d 121 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = -∞ → (-∞
< 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 116 | 115 | 2a1d 26 | . . . . . . . . . . . . . 14
⊢ (𝑦 = -∞ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 117 | 98, 110, 116 | 3jaoi 1430 | . . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 118 | 12, 117 | sylbi 217 | . . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ*
→ (∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 119 | 118 | com13 88 | . . . . . . . . . . 11
⊢ (𝐴 ⊆ ℝ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝑦 ∈ ℝ* → (-∞
< 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 120 | 119 | imp 406 | . . . . . . . . . 10
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → (𝑦 ∈ ℝ* → (-∞
< 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 121 | 120 | ralrimiv 3145 | . . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∀𝑦 ∈ ℝ* (-∞ <
𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 122 | 75, 121 | jca 511 | . . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ*
(-∞ < 𝑦 →
∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 123 |  | breq2 5147 | . . . . . . . . . . . 12
⊢ (𝑥 = -∞ → (𝑦 < 𝑥 ↔ 𝑦 < -∞)) | 
| 124 | 123 | notbid 318 | . . . . . . . . . . 11
⊢ (𝑥 = -∞ → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -∞)) | 
| 125 | 124 | ralbidv 3178 | . . . . . . . . . 10
⊢ (𝑥 = -∞ →
(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)) | 
| 126 |  | breq1 5146 | . . . . . . . . . . . 12
⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦)) | 
| 127 | 126 | imbi1d 341 | . . . . . . . . . . 11
⊢ (𝑥 = -∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 128 | 127 | ralbidv 3178 | . . . . . . . . . 10
⊢ (𝑥 = -∞ →
(∀𝑦 ∈
ℝ* (𝑥 <
𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ* (-∞ <
𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 129 | 125, 128 | anbi12d 632 | . . . . . . . . 9
⊢ (𝑥 = -∞ →
((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ*
(-∞ < 𝑦 →
∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))) | 
| 130 | 129 | rspcev 3622 | . . . . . . . 8
⊢
((-∞ ∈ ℝ* ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ*
(-∞ < 𝑦 →
∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 131 | 68, 122, 130 | sylancr 587 | . . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 132 | 67, 131 | syldan 591 | . . . . . 6
⊢ ((𝐴 ⊆ ℝ ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 133 | 132 | adantlr 715 | . . . . 5
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 134 | 50, 133 | pm2.61dan 813 | . . . 4
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) →
∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 135 |  | pnfxr 11315 | . . . . . 6
⊢ +∞
∈ ℝ* | 
| 136 |  | ral0 4513 | . . . . . . 7
⊢
∀𝑦 ∈
∅ ¬ 𝑦 <
+∞ | 
| 137 |  | pnfnlt 13170 | . . . . . . . . 9
⊢ (𝑦 ∈ ℝ*
→ ¬ +∞ < 𝑦) | 
| 138 | 137 | pm2.21d 121 | . . . . . . . 8
⊢ (𝑦 ∈ ℝ*
→ (+∞ < 𝑦
→ ∃𝑧 ∈
∅ 𝑧 < 𝑦)) | 
| 139 | 138 | rgen 3063 | . . . . . . 7
⊢
∀𝑦 ∈
ℝ* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) | 
| 140 | 136, 139 | pm3.2i 470 | . . . . . 6
⊢
(∀𝑦 ∈
∅ ¬ 𝑦 <
+∞ ∧ ∀𝑦
∈ ℝ* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) | 
| 141 |  | breq2 5147 | . . . . . . . . . 10
⊢ (𝑥 = +∞ → (𝑦 < 𝑥 ↔ 𝑦 < +∞)) | 
| 142 | 141 | notbid 318 | . . . . . . . . 9
⊢ (𝑥 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < +∞)) | 
| 143 | 142 | ralbidv 3178 | . . . . . . . 8
⊢ (𝑥 = +∞ →
(∀𝑦 ∈ ∅
¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ ∅ ¬ 𝑦 <
+∞)) | 
| 144 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦)) | 
| 145 | 144 | imbi1d 341 | . . . . . . . . 9
⊢ (𝑥 = +∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) ↔ (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) | 
| 146 | 145 | ralbidv 3178 | . . . . . . . 8
⊢ (𝑥 = +∞ →
(∀𝑦 ∈
ℝ* (𝑥 <
𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ* (+∞ <
𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) | 
| 147 | 143, 146 | anbi12d 632 | . . . . . . 7
⊢ (𝑥 = +∞ →
((∀𝑦 ∈ ∅
¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ*
(𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ ∅ ¬ 𝑦 < +∞ ∧ ∀𝑦 ∈ ℝ*
(+∞ < 𝑦 →
∃𝑧 ∈ ∅
𝑧 < 𝑦)))) | 
| 148 | 147 | rspcev 3622 | . . . . . 6
⊢
((+∞ ∈ ℝ* ∧ (∀𝑦 ∈ ∅ ¬ 𝑦 < +∞ ∧ ∀𝑦 ∈ ℝ*
(+∞ < 𝑦 →
∃𝑧 ∈ ∅
𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) | 
| 149 | 135, 140,
148 | mp2an 692 | . . . . 5
⊢
∃𝑥 ∈
ℝ* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) | 
| 150 | 149 | a1i 11 | . . . 4
⊢ (𝐴 ⊆ ℝ →
∃𝑥 ∈
ℝ* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) | 
| 151 | 6, 134, 150 | pm2.61ne 3027 | . . 3
⊢ (𝐴 ⊆ ℝ →
∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 152 | 151 | adantl 481 | . 2
⊢ ((𝐴 ⊆ ℝ*
∧ 𝐴 ⊆ ℝ)
→ ∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 153 |  | ssel 3977 | . . . . . 6
⊢ (𝐴 ⊆ ℝ*
→ (𝑦 ∈ 𝐴 → 𝑦 ∈
ℝ*)) | 
| 154 | 153, 71 | syl6 35 | . . . . 5
⊢ (𝐴 ⊆ ℝ*
→ (𝑦 ∈ 𝐴 → ¬ 𝑦 < -∞)) | 
| 155 | 154 | ralrimiv 3145 | . . . 4
⊢ (𝐴 ⊆ ℝ*
→ ∀𝑦 ∈
𝐴 ¬ 𝑦 < -∞) | 
| 156 |  | breq1 5146 | . . . . . . 7
⊢ (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦)) | 
| 157 | 156 | rspcev 3622 | . . . . . 6
⊢
((-∞ ∈ 𝐴
∧ -∞ < 𝑦)
→ ∃𝑧 ∈
𝐴 𝑧 < 𝑦) | 
| 158 | 157 | ex 412 | . . . . 5
⊢ (-∞
∈ 𝐴 → (-∞
< 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 159 | 158 | ralrimivw 3150 | . . . 4
⊢ (-∞
∈ 𝐴 →
∀𝑦 ∈
ℝ* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) | 
| 160 | 155, 159 | anim12i 613 | . . 3
⊢ ((𝐴 ⊆ ℝ*
∧ -∞ ∈ 𝐴)
→ (∀𝑦 ∈
𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ*
(-∞ < 𝑦 →
∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 161 | 68, 160, 130 | sylancr 587 | . 2
⊢ ((𝐴 ⊆ ℝ*
∧ -∞ ∈ 𝐴)
→ ∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | 
| 162 | 152, 161 | jaodan 960 | 1
⊢ ((𝐴 ⊆ ℝ*
∧ (𝐴 ⊆ ℝ
∨ -∞ ∈ 𝐴))
→ ∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) |