Step |
Hyp |
Ref |
Expression |
1 |
|
difeq2 |
⊢ ( 𝑥 = ∅ → ( 𝐴 ∖ 𝑥 ) = ( 𝐴 ∖ ∅ ) ) |
2 |
|
dif0 |
⊢ ( 𝐴 ∖ ∅ ) = 𝐴 |
3 |
1 2
|
eqtrdi |
⊢ ( 𝑥 = ∅ → ( 𝐴 ∖ 𝑥 ) = 𝐴 ) |
4 |
3
|
breq1d |
⊢ ( 𝑥 = ∅ → ( ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ↔ 𝐴 ≈ 𝐴 ) ) |
5 |
4
|
imbi2d |
⊢ ( 𝑥 = ∅ → ( ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ) ↔ ( ω ≼ 𝐴 → 𝐴 ≈ 𝐴 ) ) ) |
6 |
|
difeq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 ∖ 𝑥 ) = ( 𝐴 ∖ 𝑦 ) ) |
7 |
6
|
breq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ↔ ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) ) |
8 |
7
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ) ↔ ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) ) ) |
9 |
|
difeq2 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐴 ∖ 𝑥 ) = ( 𝐴 ∖ ( 𝑦 ∪ { 𝑧 } ) ) ) |
10 |
|
difun1 |
⊢ ( 𝐴 ∖ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) |
11 |
9 10
|
eqtrdi |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐴 ∖ 𝑥 ) = ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ) |
12 |
11
|
breq1d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ↔ ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ) ↔ ( ω ≼ 𝐴 → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) ) ) |
14 |
|
difeq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∖ 𝑥 ) = ( 𝐴 ∖ 𝐵 ) ) |
15 |
14
|
breq1d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ↔ ( 𝐴 ∖ 𝐵 ) ≈ 𝐴 ) ) |
16 |
15
|
imbi2d |
⊢ ( 𝑥 = 𝐵 → ( ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑥 ) ≈ 𝐴 ) ↔ ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝐵 ) ≈ 𝐴 ) ) ) |
17 |
|
reldom |
⊢ Rel ≼ |
18 |
17
|
brrelex2i |
⊢ ( ω ≼ 𝐴 → 𝐴 ∈ V ) |
19 |
|
enrefg |
⊢ ( 𝐴 ∈ V → 𝐴 ≈ 𝐴 ) |
20 |
18 19
|
syl |
⊢ ( ω ≼ 𝐴 → 𝐴 ≈ 𝐴 ) |
21 |
|
domen2 |
⊢ ( ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 → ( ω ≼ ( 𝐴 ∖ 𝑦 ) ↔ ω ≼ 𝐴 ) ) |
22 |
21
|
biimparc |
⊢ ( ( ω ≼ 𝐴 ∧ ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) → ω ≼ ( 𝐴 ∖ 𝑦 ) ) |
23 |
|
infdifsn |
⊢ ( ω ≼ ( 𝐴 ∖ 𝑦 ) → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ ( 𝐴 ∖ 𝑦 ) ) |
24 |
22 23
|
syl |
⊢ ( ( ω ≼ 𝐴 ∧ ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ ( 𝐴 ∖ 𝑦 ) ) |
25 |
|
entr |
⊢ ( ( ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ ( 𝐴 ∖ 𝑦 ) ∧ ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) |
26 |
24 25
|
sylancom |
⊢ ( ( ω ≼ 𝐴 ∧ ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) |
27 |
26
|
ex |
⊢ ( ω ≼ 𝐴 → ( ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) ) |
28 |
27
|
a2i |
⊢ ( ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) → ( ω ≼ 𝐴 → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) ) |
29 |
28
|
a1i |
⊢ ( 𝑦 ∈ Fin → ( ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝑦 ) ≈ 𝐴 ) → ( ω ≼ 𝐴 → ( ( 𝐴 ∖ 𝑦 ) ∖ { 𝑧 } ) ≈ 𝐴 ) ) ) |
30 |
5 8 13 16 20 29
|
findcard2 |
⊢ ( 𝐵 ∈ Fin → ( ω ≼ 𝐴 → ( 𝐴 ∖ 𝐵 ) ≈ 𝐴 ) ) |
31 |
30
|
impcom |
⊢ ( ( ω ≼ 𝐴 ∧ 𝐵 ∈ Fin ) → ( 𝐴 ∖ 𝐵 ) ≈ 𝐴 ) |