Proof of Theorem 2ndcdisj2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-rmo 3379 | . . 3
⊢
(∃*𝑥 ∈
𝐴 𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | 
| 2 | 1 | albii 1818 | . 2
⊢
(∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | 
| 3 |  | undif2 4476 | . . . . . 6
⊢
({∅} ∪ (𝐴
∖ {∅})) = ({∅} ∪ 𝐴) | 
| 4 |  | omex 9684 | . . . . . . . 8
⊢ ω
∈ V | 
| 5 |  | peano1 7911 | . . . . . . . . 9
⊢ ∅
∈ ω | 
| 6 |  | snssi 4807 | . . . . . . . . 9
⊢ (∅
∈ ω → {∅} ⊆ ω) | 
| 7 | 5, 6 | ax-mp 5 | . . . . . . . 8
⊢ {∅}
⊆ ω | 
| 8 |  | ssdomg 9041 | . . . . . . . 8
⊢ (ω
∈ V → ({∅} ⊆ ω → {∅} ≼
ω)) | 
| 9 | 4, 7, 8 | mp2 9 | . . . . . . 7
⊢ {∅}
≼ ω | 
| 10 |  | id 22 | . . . . . . . 8
⊢ (𝐽 ∈ 2ndω
→ 𝐽 ∈
2ndω) | 
| 11 |  | ssdif 4143 | . . . . . . . . 9
⊢ (𝐴 ⊆ 𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖
{∅})) | 
| 12 |  | dfss3 3971 | . . . . . . . . 9
⊢ ((𝐴 ∖ {∅}) ⊆
(𝐽 ∖ {∅})
↔ ∀𝑥 ∈
(𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅})) | 
| 13 | 11, 12 | sylib 218 | . . . . . . . 8
⊢ (𝐴 ⊆ 𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅})) | 
| 14 |  | eldifi 4130 | . . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥 ∈ 𝐴) | 
| 15 | 14 | anim1i 615 | . . . . . . . . . 10
⊢ ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | 
| 16 | 15 | moimi 2544 | . . . . . . . . 9
⊢
(∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) | 
| 17 | 16 | alimi 1810 | . . . . . . . 8
⊢
(∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) | 
| 18 |  | df-rmo 3379 | . . . . . . . . . 10
⊢
(∃*𝑥 ∈
(𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) | 
| 19 | 18 | albii 1818 | . . . . . . . . 9
⊢
(∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) | 
| 20 |  | 2ndcdisj 23465 | . . . . . . . . 9
⊢ ((𝐽 ∈ 2ndω
∧ ∀𝑥 ∈
(𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥) → (𝐴 ∖ {∅}) ≼
ω) | 
| 21 | 19, 20 | syl3an3br 1409 | . . . . . . . 8
⊢ ((𝐽 ∈ 2ndω
∧ ∀𝑥 ∈
(𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼
ω) | 
| 22 | 10, 13, 17, 21 | syl3an 1160 | . . . . . . 7
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼
ω) | 
| 23 |  | unctb 10245 | . . . . . . 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 5178 | . . . . 5
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ≼ ω) | 
| 26 |  | ctex 9005 | . . . . 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 4178 | . . . 4
⊢ 𝐴 ⊆ ({∅} ∪ 𝐴) | 
| 29 |  | ssdomg 9041 | . . . 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 9048 | . . 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 1406 | 1
⊢ ((𝐽 ∈ 2ndω
∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → 𝐴 ≼ ω) |