Step | Hyp | Ref
| Expression |
1 | | ssrab2 3213 |
. . . 4
⊢ {𝑤 ∈ {∅, {∅}}
∣ (𝑤 = {∅} ∨
(𝑤 = ∅ ∧ 𝜑))} ⊆ {∅,
{∅}} |
2 | | peano1 4553 |
. . . . 5
⊢ ∅
∈ ω |
3 | | suc0 4371 |
. . . . . 6
⊢ suc
∅ = {∅} |
4 | | peano2 4554 |
. . . . . . 7
⊢ (∅
∈ ω → suc ∅ ∈ ω) |
5 | 2, 4 | ax-mp 5 |
. . . . . 6
⊢ suc
∅ ∈ ω |
6 | 3, 5 | eqeltrri 2231 |
. . . . 5
⊢ {∅}
∈ ω |
7 | | prssi 3714 |
. . . . 5
⊢ ((∅
∈ ω ∧ {∅} ∈ ω) → {∅, {∅}}
⊆ ω) |
8 | 2, 6, 7 | mp2an 423 |
. . . 4
⊢ {∅,
{∅}} ⊆ ω |
9 | 1, 8 | sstri 3137 |
. . 3
⊢ {𝑤 ∈ {∅, {∅}}
∣ (𝑤 = {∅} ∨
(𝑤 = ∅ ∧ 𝜑))} ⊆
ω |
10 | | eqid 2157 |
. . . 4
⊢ {𝑤 ∈ {∅, {∅}}
∣ (𝑤 = {∅} ∨
(𝑤 = ∅ ∧ 𝜑))} = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} |
11 | 10 | regexmidlemm 4491 |
. . 3
⊢
∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} |
12 | | pp0ex 4150 |
. . . . 5
⊢ {∅,
{∅}} ∈ V |
13 | 12 | rabex 4108 |
. . . 4
⊢ {𝑤 ∈ {∅, {∅}}
∣ (𝑤 = {∅} ∨
(𝑤 = ∅ ∧ 𝜑))} ∈ V |
14 | | sseq1 3151 |
. . . . . 6
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑥 ⊆ ω ↔ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω)) |
15 | | eleq2 2221 |
. . . . . . 7
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) |
16 | 15 | exbidv 1805 |
. . . . . 6
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) |
17 | 14, 16 | anbi12d 465 |
. . . . 5
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) ↔ ({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))) |
18 | | eleq2 2221 |
. . . . . . . . . 10
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) |
19 | 18 | notbid 657 |
. . . . . . . . 9
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) |
20 | 19 | imbi2d 229 |
. . . . . . . 8
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))) |
21 | 20 | albidv 1804 |
. . . . . . 7
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))) |
22 | 15, 21 | anbi12d 465 |
. . . . . 6
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))) |
23 | 22 | exbidv 1805 |
. . . . 5
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) ↔ ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))) |
24 | 17, 23 | imbi12d 233 |
. . . 4
⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ (({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}) → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))) |
25 | | nnregexmid.1 |
. . . 4
⊢ ((𝑥 ⊆ ω ∧
∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
26 | 13, 24, 25 | vtocl 2766 |
. . 3
⊢ (({𝑤 ∈ {∅, {∅}}
∣ (𝑤 = {∅} ∨
(𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧
∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}) → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))) |
27 | 9, 11, 26 | mp2an 423 |
. 2
⊢
∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) |
28 | 10 | regexmidlem1 4492 |
. 2
⊢
(∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) → (𝜑 ∨ ¬ 𝜑)) |
29 | 27, 28 | ax-mp 5 |
1
⊢ (𝜑 ∨ ¬ 𝜑) |