Step | Hyp | Ref
| Expression |
1 | | 2onn 8472 |
. . . . . . . . . 10
⊢
2o ∈ ω |
2 | | nnfi 8950 |
. . . . . . . . . 10
⊢
(2o ∈ ω → 2o ∈
Fin) |
3 | 1, 2 | ax-mp 5 |
. . . . . . . . 9
⊢
2o ∈ Fin |
4 | | enfi 8973 |
. . . . . . . . 9
⊢ (𝑃 ≈ 2o →
(𝑃 ∈ Fin ↔
2o ∈ Fin)) |
5 | 3, 4 | mpbiri 257 |
. . . . . . . 8
⊢ (𝑃 ≈ 2o →
𝑃 ∈
Fin) |
6 | 5 | adantl 482 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin) |
7 | | diffi 8962 |
. . . . . . 7
⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin) |
8 | 6, 7 | syl 17 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin) |
9 | 8 | cardidd 10305 |
. . . . 5
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘(𝑃 ∖
{𝑋})) ≈ (𝑃 ∖ {𝑋})) |
10 | 9 | ensymd 8791 |
. . . 4
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋}))) |
11 | | simpl 483 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃) |
12 | | dif1card 9766 |
. . . . . . 7
⊢ ((𝑃 ∈ Fin ∧ 𝑋 ∈ 𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋}))) |
13 | 6, 11, 12 | syl2anc 584 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘𝑃) = suc
(card‘(𝑃 ∖
{𝑋}))) |
14 | | cardennn 9741 |
. . . . . . . . 9
⊢ ((𝑃 ≈ 2o ∧
2o ∈ ω) → (card‘𝑃) = 2o) |
15 | 1, 14 | mpan2 688 |
. . . . . . . 8
⊢ (𝑃 ≈ 2o →
(card‘𝑃) =
2o) |
16 | | df-2o 8298 |
. . . . . . . 8
⊢
2o = suc 1o |
17 | 15, 16 | eqtrdi 2794 |
. . . . . . 7
⊢ (𝑃 ≈ 2o →
(card‘𝑃) = suc
1o) |
18 | 17 | adantl 482 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘𝑃) = suc
1o) |
19 | 13, 18 | eqtr3d 2780 |
. . . . 5
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc
(card‘(𝑃 ∖
{𝑋})) = suc
1o) |
20 | | suc11reg 9377 |
. . . . 5
⊢ (suc
(card‘(𝑃 ∖
{𝑋})) = suc 1o
↔ (card‘(𝑃
∖ {𝑋})) =
1o) |
21 | 19, 20 | sylib 217 |
. . . 4
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) →
(card‘(𝑃 ∖
{𝑋})) =
1o) |
22 | 10, 21 | breqtrd 5100 |
. . 3
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o) |
23 | | en1 8811 |
. . 3
⊢ ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥}) |
24 | 22, 23 | sylib 217 |
. 2
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥}) |
25 | | simpr 485 |
. . . . 5
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥}) |
26 | 25 | unieqd 4853 |
. . . 4
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥}) |
27 | | vex 3436 |
. . . . 5
⊢ 𝑥 ∈ V |
28 | 27 | unisn 4861 |
. . . 4
⊢ ∪ {𝑥}
= 𝑥 |
29 | 26, 28 | eqtrdi 2794 |
. . 3
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥) |
30 | | difssd 4067 |
. . . . 5
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ⊆ 𝑃) |
31 | 25, 30 | eqsstrrd 3960 |
. . . 4
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → {𝑥} ⊆ 𝑃) |
32 | | vsnid 4598 |
. . . 4
⊢ 𝑥 ∈ {𝑥} |
33 | | ssel2 3916 |
. . . 4
⊢ (({𝑥} ⊆ 𝑃 ∧ 𝑥 ∈ {𝑥}) → 𝑥 ∈ 𝑃) |
34 | 31, 32, 33 | sylancl 586 |
. . 3
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑥 ∈ 𝑃) |
35 | 29, 34 | eqeltrd 2839 |
. 2
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃) |
36 | 24, 35 | exlimddv 1938 |
1
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃
∖ {𝑋}) ∈ 𝑃) |