Proof of Theorem 2ndcdisj2
Step | Hyp | Ref
| Expression |
1 | | df-rmo 3071 |
. . 3
⊢
(∃*𝑥 ∈
𝐴 𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) |
2 | 1 | albii 1822 |
. 2
⊢
(∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) |
3 | | undif2 4410 |
. . . . . 6
⊢
({∅} ∪ (𝐴
∖ {∅})) = ({∅} ∪ 𝐴) |
4 | | omex 9401 |
. . . . . . . 8
⊢ ω
∈ V |
5 | | peano1 7735 |
. . . . . . . . 9
⊢ ∅
∈ ω |
6 | | snssi 4741 |
. . . . . . . . 9
⊢ (∅
∈ ω → {∅} ⊆ ω) |
7 | 5, 6 | ax-mp 5 |
. . . . . . . 8
⊢ {∅}
⊆ ω |
8 | | ssdomg 8786 |
. . . . . . . 8
⊢ (ω
∈ V → ({∅} ⊆ ω → {∅} ≼
ω)) |
9 | 4, 7, 8 | mp2 9 |
. . . . . . 7
⊢ {∅}
≼ ω |
10 | | id 22 |
. . . . . . . 8
⊢ (𝐽 ∈ 2ndω
→ 𝐽 ∈
2ndω) |
11 | | ssdif 4074 |
. . . . . . . . 9
⊢ (𝐴 ⊆ 𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖
{∅})) |
12 | | dfss3 3909 |
. . . . . . . . 9
⊢ ((𝐴 ∖ {∅}) ⊆
(𝐽 ∖ {∅})
↔ ∀𝑥 ∈
(𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅})) |
13 | 11, 12 | sylib 217 |
. . . . . . . 8
⊢ (𝐴 ⊆ 𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅})) |
14 | | eldifi 4061 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥 ∈ 𝐴) |
15 | 14 | anim1i 615 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) |
16 | 15 | moimi 2545 |
. . . . . . . . 9
⊢
(∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) |
17 | 16 | alimi 1814 |
. . . . . . . 8
⊢
(∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) |
18 | | df-rmo 3071 |
. . . . . . . . . 10
⊢
(∃*𝑥 ∈
(𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) |
19 | 18 | albii 1822 |
. . . . . . . . 9
⊢
(∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) |
20 | | 2ndcdisj 22607 |
. . . . . . . . 9
⊢ ((𝐽 ∈ 2ndω
∧ ∀𝑥 ∈
(𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥) → (𝐴 ∖ {∅}) ≼
ω) |
21 | 19, 20 | syl3an3br 1407 |
. . . . . . . 8
⊢ ((𝐽 ∈ 2ndω
∧ ∀𝑥 ∈
(𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼
ω) |
22 | 10, 13, 17, 21 | syl3an 1159 |
. . . . . . 7
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼
ω) |
23 | | unctb 9961 |
. . . . . . 7
⊢
(({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) →
({∅} ∪ (𝐴 ∖
{∅})) ≼ ω) |
24 | 9, 22, 23 | sylancr 587 |
. . . . . 6
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼
ω) |
25 | 3, 24 | eqbrtrrid 5110 |
. . . . 5
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ≼ ω) |
26 | | ctex 8753 |
. . . . 5
⊢
(({∅} ∪ 𝐴)
≼ ω → ({∅} ∪ 𝐴) ∈ V) |
27 | 25, 26 | syl 17 |
. . . 4
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ∈ V) |
28 | | ssun2 4107 |
. . . 4
⊢ 𝐴 ⊆ ({∅} ∪ 𝐴) |
29 | | ssdomg 8786 |
. . . 4
⊢
(({∅} ∪ 𝐴)
∈ V → (𝐴 ⊆
({∅} ∪ 𝐴) →
𝐴 ≼ ({∅} ∪
𝐴))) |
30 | 27, 28, 29 | mpisyl 21 |
. . 3
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝐴 ≼ ({∅} ∪ 𝐴)) |
31 | | domtr 8793 |
. . 3
⊢ ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼
ω) |
32 | 30, 25, 31 | syl2anc 584 |
. 2
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝐴 ≼ ω) |
33 | 2, 32 | syl3an3b 1404 |
1
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → 𝐴 ≼ ω) |