| Step | Hyp | Ref
| Expression |
| 1 | | ssrab2 4080 |
. . . . . 6
⊢ {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ 𝑆 |
| 2 | 1 | a1i 11 |
. . . . 5
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ 𝑆) |
| 3 | | id 22 |
. . . . 5
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → 𝑆 ⊆ (ℤ≥‘𝑀)) |
| 4 | 2, 3 | sstrd 3994 |
. . . 4
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆
(ℤ≥‘𝑀)) |
| 5 | 4 | adantr 480 |
. . 3
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆
(ℤ≥‘𝑀)) |
| 6 | | rabn0 4389 |
. . . . . 6
⊢ ({𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅ ↔ ∃𝑗 ∈ 𝑆 𝜑) |
| 7 | 6 | bicomi 224 |
. . . . 5
⊢
(∃𝑗 ∈
𝑆 𝜑 ↔ {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅) |
| 8 | 7 | biimpi 216 |
. . . 4
⊢
(∃𝑗 ∈
𝑆 𝜑 → {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅) |
| 9 | 8 | adantl 481 |
. . 3
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅) |
| 10 | | uzwo 12953 |
. . 3
⊢ (({𝑗 ∈ 𝑆 ∣ 𝜑} ⊆
(ℤ≥‘𝑀) ∧ {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅) → ∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) |
| 11 | 5, 9, 10 | syl2anc 584 |
. 2
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) |
| 12 | 1 | sseli 3979 |
. . . . . . . 8
⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → 𝑖 ∈ 𝑆) |
| 13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ 𝑆) |
| 14 | 13 | 3adant1 1131 |
. . . . . 6
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ 𝑆) |
| 15 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝑖 |
| 16 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝑆 |
| 17 | 15 | nfsbc1 3807 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗[𝑖 / 𝑗]𝜑 |
| 18 | | sbceq1a 3799 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → (𝜑 ↔ [𝑖 / 𝑗]𝜑)) |
| 19 | 15, 16, 17, 18 | elrabf 3688 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ↔ (𝑖 ∈ 𝑆 ∧ [𝑖 / 𝑗]𝜑)) |
| 20 | 19 | biimpi 216 |
. . . . . . . . . 10
⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → (𝑖 ∈ 𝑆 ∧ [𝑖 / 𝑗]𝜑)) |
| 21 | 20 | simprd 495 |
. . . . . . . . 9
⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → [𝑖 / 𝑗]𝜑) |
| 22 | 21 | adantr 480 |
. . . . . . . 8
⊢ ((𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → [𝑖 / 𝑗]𝜑) |
| 23 | 22 | 3adant1 1131 |
. . . . . . 7
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → [𝑖 / 𝑗]𝜑) |
| 24 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑘 𝑆 ⊆
(ℤ≥‘𝑀) |
| 25 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑘 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} |
| 26 | | nfra1 3284 |
. . . . . . . . 9
⊢
Ⅎ𝑘∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 |
| 27 | 24, 25, 26 | nf3an 1901 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) |
| 28 | | simpl13 1251 |
. . . . . . . . . . 11
⊢ ((((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) |
| 29 | | simpl2 1193 |
. . . . . . . . . . 11
⊢ ((((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝑘 ∈ 𝑆) |
| 30 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝜓) |
| 31 | | simpll 767 |
. . . . . . . . . . . 12
⊢
(((∀𝑘 ∈
{𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) |
| 32 | | id 22 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ 𝑆 ∧ 𝜓) → (𝑘 ∈ 𝑆 ∧ 𝜓)) |
| 33 | | nfcv 2905 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝑘 |
| 34 | | uzwo4.1 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜓 |
| 35 | | uzwo4.2 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓)) |
| 36 | 33, 16, 34, 35 | elrabf 3688 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ↔ (𝑘 ∈ 𝑆 ∧ 𝜓)) |
| 37 | 32, 36 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝑆 ∧ 𝜓) → 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) |
| 38 | 37 | adantll 714 |
. . . . . . . . . . . 12
⊢
(((∀𝑘 ∈
{𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) |
| 39 | | rspa 3248 |
. . . . . . . . . . . 12
⊢
((∀𝑘 ∈
{𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ≤ 𝑘) |
| 40 | 31, 38, 39 | syl2anc 584 |
. . . . . . . . . . 11
⊢
(((∀𝑘 ∈
{𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → 𝑖 ≤ 𝑘) |
| 41 | 28, 29, 30, 40 | syl21anc 838 |
. . . . . . . . . 10
⊢ ((((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝑖 ≤ 𝑘) |
| 42 | 4 | sselda 3983 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ (ℤ≥‘𝑀)) |
| 43 | | eluzelz 12888 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈
(ℤ≥‘𝑀) → 𝑖 ∈ ℤ) |
| 44 | 42, 43 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ ℤ) |
| 45 | 44 | zred 12722 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ ℝ) |
| 46 | 45 | 3adant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ ℝ) |
| 47 | 46 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ (((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑖 ∈ ℝ) |
| 48 | | ssel2 3978 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 49 | | eluzelz 12888 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → 𝑘 ∈ ℤ) |
| 50 | 48, 49 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℤ) |
| 51 | 50 | zred 12722 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℝ) |
| 52 | 51 | 3ad2antl1 1186 |
. . . . . . . . . . . . 13
⊢ (((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℝ) |
| 53 | 52 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ (((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑘 ∈ ℝ) |
| 54 | | simp3 1139 |
. . . . . . . . . . . 12
⊢ (((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑘 < 𝑖) |
| 55 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑘 < 𝑖) |
| 56 | | simp2 1138 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑘 ∈ ℝ) |
| 57 | | simp1 1137 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑖 ∈ ℝ) |
| 58 | 56, 57 | ltnled 11408 |
. . . . . . . . . . . . 13
⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → (𝑘 < 𝑖 ↔ ¬ 𝑖 ≤ 𝑘)) |
| 59 | 55, 58 | mpbid 232 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → ¬ 𝑖 ≤ 𝑘) |
| 60 | 47, 53, 54, 59 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → ¬ 𝑖 ≤ 𝑘) |
| 61 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → ¬ 𝑖 ≤ 𝑘) |
| 62 | 41, 61 | pm2.65da 817 |
. . . . . . . . 9
⊢ (((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → ¬ 𝜓) |
| 63 | 62 | 3exp 1120 |
. . . . . . . 8
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → (𝑘 ∈ 𝑆 → (𝑘 < 𝑖 → ¬ 𝜓))) |
| 64 | 27, 63 | ralrimi 3257 |
. . . . . . 7
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)) |
| 65 | 23, 64 | jca 511 |
. . . . . 6
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))) |
| 66 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 𝑘 < 𝑖 |
| 67 | 34 | nfn 1857 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 ¬ 𝜓 |
| 68 | 66, 67 | nfim 1896 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝑘 < 𝑖 → ¬ 𝜓) |
| 69 | 16, 68 | nfralw 3311 |
. . . . . . . 8
⊢
Ⅎ𝑗∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓) |
| 70 | 17, 69 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑗([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)) |
| 71 | | breq2 5147 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → (𝑘 < 𝑗 ↔ 𝑘 < 𝑖)) |
| 72 | 71 | imbi1d 341 |
. . . . . . . . 9
⊢ (𝑗 = 𝑖 → ((𝑘 < 𝑗 → ¬ 𝜓) ↔ (𝑘 < 𝑖 → ¬ 𝜓))) |
| 73 | 72 | ralbidv 3178 |
. . . . . . . 8
⊢ (𝑗 = 𝑖 → (∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓) ↔ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))) |
| 74 | 18, 73 | anbi12d 632 |
. . . . . . 7
⊢ (𝑗 = 𝑖 → ((𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)) ↔ ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)))) |
| 75 | 70, 74 | rspce 3611 |
. . . . . 6
⊢ ((𝑖 ∈ 𝑆 ∧ ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))) |
| 76 | 14, 65, 75 | syl2anc 584 |
. . . . 5
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))) |
| 77 | 76 | 3exp 1120 |
. . . 4
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → (∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))))) |
| 78 | 77 | rexlimdv 3153 |
. . 3
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))) |
| 79 | 78 | adantr 480 |
. 2
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → (∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))) |
| 80 | 11, 79 | mpd 15 |
1
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))) |