Step | Hyp | Ref
| Expression |
1 | | isf32lem.a |
. . . 4
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) |
2 | | isf32lem.b |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) |
3 | | isf32lem.c |
. . . 4
⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) |
4 | 1, 2, 3 | isf32lem2 9176 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
5 | 4 | ralrimiva 2966 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
6 | | isf32lem.d |
. . . . . . . 8
⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} |
7 | | ssrab2 3687 |
. . . . . . . 8
⊢ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ⊆ ω |
8 | 6, 7 | eqsstri 3635 |
. . . . . . 7
⊢ 𝑆 ⊆
ω |
9 | | nnunifi 8211 |
. . . . . . 7
⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆
∈ ω) |
10 | 8, 9 | mpan 706 |
. . . . . 6
⊢ (𝑆 ∈ Fin → ∪ 𝑆
∈ ω) |
11 | 10 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆
∈ ω) |
12 | | elssuni 4467 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ 𝑆 → 𝑏 ⊆ ∪ 𝑆) |
13 | | nnon 7071 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ ω → 𝑏 ∈ On) |
14 | | omsson 7069 |
. . . . . . . . . . . . . . 15
⊢ ω
⊆ On |
15 | 14, 11 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆
∈ On) |
16 | | ontri1 5757 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 ∈ On ∧ ∪ 𝑆
∈ On) → (𝑏
⊆ ∪ 𝑆 ↔ ¬ ∪
𝑆 ∈ 𝑏)) |
17 | 13, 15, 16 | syl2anr 495 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆
∈ 𝑏)) |
18 | 12, 17 | syl5ib 234 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ∈ 𝑆 → ¬ ∪
𝑆 ∈ 𝑏)) |
19 | 18 | con2d 129 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆
∈ 𝑏 → ¬ 𝑏 ∈ 𝑆)) |
20 | 19 | impr 649 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆
∈ 𝑏)) → ¬
𝑏 ∈ 𝑆) |
21 | 6 | eleq2i 2693 |
. . . . . . . . . 10
⊢ (𝑏 ∈ 𝑆 ↔ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}) |
22 | 20, 21 | sylnib 318 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆
∈ 𝑏)) → ¬
𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}) |
23 | | suceq 5790 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏) |
24 | 23 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑏 → (𝐹‘suc 𝑦) = (𝐹‘suc 𝑏)) |
25 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏)) |
26 | 24, 25 | psseq12d 3701 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑏 → ((𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦) ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
27 | 26 | elrab3 3364 |
. . . . . . . . . 10
⊢ (𝑏 ∈ ω → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
28 | 27 | ad2antrl 764 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆
∈ 𝑏)) → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
29 | 22, 28 | mtbid 314 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆
∈ 𝑏)) → ¬
(𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) |
30 | 29 | expr 643 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆
∈ 𝑏 → ¬
(𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
31 | | imnan 438 |
. . . . . . 7
⊢ ((∪ 𝑆
∈ 𝑏 → ¬
(𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ (∪
𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
32 | 30, 31 | sylib 208 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → ¬ (∪ 𝑆
∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
33 | 32 | nrexdv 3001 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∃𝑏 ∈ ω (∪ 𝑆
∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
34 | | eleq1 2689 |
. . . . . . . . 9
⊢ (𝑎 = ∪
𝑆 → (𝑎 ∈ 𝑏 ↔ ∪ 𝑆 ∈ 𝑏)) |
35 | 34 | anbi1d 741 |
. . . . . . . 8
⊢ (𝑎 = ∪
𝑆 → ((𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))) |
36 | 35 | rexbidv 3052 |
. . . . . . 7
⊢ (𝑎 = ∪
𝑆 → (∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ∃𝑏 ∈ ω (∪
𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))) |
37 | 36 | notbid 308 |
. . . . . 6
⊢ (𝑎 = ∪
𝑆 → (¬
∃𝑏 ∈ ω
(𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ ∃𝑏 ∈ ω (∪
𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))) |
38 | 37 | rspcev 3309 |
. . . . 5
⊢ ((∪ 𝑆
∈ ω ∧ ¬ ∃𝑏 ∈ ω (∪
𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) → ∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
39 | 11, 33, 38 | syl2anc 693 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∃𝑎 ∈ ω ¬
∃𝑏 ∈ ω
(𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
40 | | rexnal 2995 |
. . . 4
⊢
(∃𝑎 ∈
ω ¬ ∃𝑏
∈ ω (𝑎 ∈
𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
41 | 39, 40 | sylib 208 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) |
42 | 41 | ex 450 |
. 2
⊢ (𝜑 → (𝑆 ∈ Fin → ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))) |
43 | 5, 42 | mt2d 131 |
1
⊢ (𝜑 → ¬ 𝑆 ∈ Fin) |