Step | Hyp | Ref
| Expression |
1 | | 2onn 8297 |
. . . . . . . . . 10
⊢
2o ∈ ω |
2 | | nnfi 8766 |
. . . . . . . . . 10
⊢
(2o ∈ ω → 2o ∈
Fin) |
3 | 1, 2 | ax-mp 5 |
. . . . . . . . 9
⊢
2o ∈ Fin |
4 | | enfi 8785 |
. . . . . . . . 9
⊢ (𝑃 ≈ 2o →
(𝑃 ∈ Fin ↔
2o ∈ Fin)) |
5 | 3, 4 | mpbiri 261 |
. . . . . . . 8
⊢ (𝑃 ≈ 2o →
𝑃 ∈
Fin) |
6 | 5 | adantl 485 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin) |
7 | | diffi 8827 |
. . . . . . 7
⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin) |
8 | 6, 7 | syl 17 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin) |
9 | 8 | cardidd 10049 |
. . . . 5
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘(𝑃 ∖
{𝑋})) ≈ (𝑃 ∖ {𝑋})) |
10 | 9 | ensymd 8606 |
. . . 4
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋}))) |
11 | | simpl 486 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃) |
12 | | dif1card 9510 |
. . . . . . 7
⊢ ((𝑃 ∈ Fin ∧ 𝑋 ∈ 𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋}))) |
13 | 6, 11, 12 | syl2anc 587 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘𝑃) = suc
(card‘(𝑃 ∖
{𝑋}))) |
14 | | cardennn 9485 |
. . . . . . . . 9
⊢ ((𝑃 ≈ 2o ∧
2o ∈ ω) → (card‘𝑃) = 2o) |
15 | 1, 14 | mpan2 691 |
. . . . . . . 8
⊢ (𝑃 ≈ 2o →
(card‘𝑃) =
2o) |
16 | | df-2o 8132 |
. . . . . . . 8
⊢
2o = suc 1o |
17 | 15, 16 | eqtrdi 2789 |
. . . . . . 7
⊢ (𝑃 ≈ 2o →
(card‘𝑃) = suc
1o) |
18 | 17 | adantl 485 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘𝑃) = suc
1o) |
19 | 13, 18 | eqtr3d 2775 |
. . . . 5
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc
(card‘(𝑃 ∖
{𝑋})) = suc
1o) |
20 | | suc11reg 9155 |
. . . . 5
⊢ (suc
(card‘(𝑃 ∖
{𝑋})) = suc 1o
↔ (card‘(𝑃
∖ {𝑋})) =
1o) |
21 | 19, 20 | sylib 221 |
. . . 4
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘(𝑃 ∖
{𝑋})) =
1o) |
22 | 10, 21 | breqtrd 5056 |
. . 3
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o) |
23 | | en1 8623 |
. . 3
⊢ ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥}) |
24 | 22, 23 | sylib 221 |
. 2
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥}) |
25 | | simplll 775 |
. . . . . . 7
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ 𝑃) |
26 | 25 | elexd 3418 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ V) |
27 | | simplr 769 |
. . . . . . . . . . 11
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑥}) |
28 | | sneqbg 4729 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ 𝑃 → ({𝑋} = {𝑥} ↔ 𝑋 = 𝑥)) |
29 | 28 | biimpar 481 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥}) |
30 | 29 | ad4ant14 752 |
. . . . . . . . . . 11
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥}) |
31 | 27, 30 | eqtr4d 2776 |
. . . . . . . . . 10
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋}) |
32 | 31 | ineq2d 4103 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋})) |
33 | | disjdif 4361 |
. . . . . . . . 9
⊢ ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅ |
34 | | inidm 4109 |
. . . . . . . . 9
⊢ ({𝑋} ∩ {𝑋}) = {𝑋} |
35 | 32, 33, 34 | 3eqtr3g 2796 |
. . . . . . . 8
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋}) |
36 | 35 | eqcomd 2744 |
. . . . . . 7
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅) |
37 | | snprc 4608 |
. . . . . . 7
⊢ (¬
𝑋 ∈ V ↔ {𝑋} = ∅) |
38 | 36, 37 | sylibr 237 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V) |
39 | 26, 38 | pm2.65da 817 |
. . . . 5
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥) |
40 | 39 | neqned 2941 |
. . . 4
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 ≠ 𝑥) |
41 | | simpr 488 |
. . . . . 6
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥}) |
42 | 41 | unieqd 4810 |
. . . . 5
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥}) |
43 | | vex 3402 |
. . . . . 6
⊢ 𝑥 ∈ V |
44 | 43 | unisn 4818 |
. . . . 5
⊢ ∪ {𝑥}
= 𝑥 |
45 | 42, 44 | eqtrdi 2789 |
. . . 4
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥) |
46 | 40, 45 | neeqtrrd 3008 |
. . 3
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) |
47 | 46 | necomd 2989 |
. 2
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋) |
48 | 24, 47 | exlimddv 1942 |
1
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃
∖ {𝑋}) ≠ 𝑋) |