Step |
Hyp |
Ref |
Expression |
1 |
|
encv |
⊢ ( 𝐴 ≈ suc 𝑀 → ( 𝐴 ∈ V ∧ suc 𝑀 ∈ V ) ) |
2 |
1
|
simpld |
⊢ ( 𝐴 ≈ suc 𝑀 → 𝐴 ∈ V ) |
3 |
|
breng |
⊢ ( ( 𝐴 ∈ V ∧ suc 𝑀 ∈ V ) → ( 𝐴 ≈ suc 𝑀 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) ) |
4 |
1 3
|
syl |
⊢ ( 𝐴 ≈ suc 𝑀 → ( 𝐴 ≈ suc 𝑀 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) ) |
5 |
4
|
ibi |
⊢ ( 𝐴 ≈ suc 𝑀 → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) |
6 |
|
sucidg |
⊢ ( 𝑀 ∈ On → 𝑀 ∈ suc 𝑀 ) |
7 |
|
f1ocnvdm |
⊢ ( ( 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ∧ 𝑀 ∈ suc 𝑀 ) → ( ◡ 𝑓 ‘ 𝑀 ) ∈ 𝐴 ) |
8 |
7
|
ancoms |
⊢ ( ( 𝑀 ∈ suc 𝑀 ∧ 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) → ( ◡ 𝑓 ‘ 𝑀 ) ∈ 𝐴 ) |
9 |
6 8
|
sylan |
⊢ ( ( 𝑀 ∈ On ∧ 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) → ( ◡ 𝑓 ‘ 𝑀 ) ∈ 𝐴 ) |
10 |
9
|
adantll |
⊢ ( ( ( 𝐴 ∈ V ∧ 𝑀 ∈ On ) ∧ 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) → ( ◡ 𝑓 ‘ 𝑀 ) ∈ 𝐴 ) |
11 |
|
vex |
⊢ 𝑓 ∈ V |
12 |
|
dif1enlem |
⊢ ( ( ( 𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On ) ∧ 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) → ( 𝐴 ∖ { ( ◡ 𝑓 ‘ 𝑀 ) } ) ≈ 𝑀 ) |
13 |
11 12
|
mp3anl1 |
⊢ ( ( ( 𝐴 ∈ V ∧ 𝑀 ∈ On ) ∧ 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) → ( 𝐴 ∖ { ( ◡ 𝑓 ‘ 𝑀 ) } ) ≈ 𝑀 ) |
14 |
|
sneq |
⊢ ( 𝑥 = ( ◡ 𝑓 ‘ 𝑀 ) → { 𝑥 } = { ( ◡ 𝑓 ‘ 𝑀 ) } ) |
15 |
14
|
difeq2d |
⊢ ( 𝑥 = ( ◡ 𝑓 ‘ 𝑀 ) → ( 𝐴 ∖ { 𝑥 } ) = ( 𝐴 ∖ { ( ◡ 𝑓 ‘ 𝑀 ) } ) ) |
16 |
15
|
breq1d |
⊢ ( 𝑥 = ( ◡ 𝑓 ‘ 𝑀 ) → ( ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ↔ ( 𝐴 ∖ { ( ◡ 𝑓 ‘ 𝑀 ) } ) ≈ 𝑀 ) ) |
17 |
16
|
rspcev |
⊢ ( ( ( ◡ 𝑓 ‘ 𝑀 ) ∈ 𝐴 ∧ ( 𝐴 ∖ { ( ◡ 𝑓 ‘ 𝑀 ) } ) ≈ 𝑀 ) → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) |
18 |
10 13 17
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ V ∧ 𝑀 ∈ On ) ∧ 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 ) → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) |
19 |
18
|
ex |
⊢ ( ( 𝐴 ∈ V ∧ 𝑀 ∈ On ) → ( 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) ) |
20 |
19
|
exlimdv |
⊢ ( ( 𝐴 ∈ V ∧ 𝑀 ∈ On ) → ( ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ suc 𝑀 → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) ) |
21 |
5 20
|
syl5 |
⊢ ( ( 𝐴 ∈ V ∧ 𝑀 ∈ On ) → ( 𝐴 ≈ suc 𝑀 → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) ) |
22 |
2 21
|
sylan |
⊢ ( ( 𝐴 ≈ suc 𝑀 ∧ 𝑀 ∈ On ) → ( 𝐴 ≈ suc 𝑀 → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) ) |
23 |
22
|
ancoms |
⊢ ( ( 𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ) → ( 𝐴 ≈ suc 𝑀 → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) ) |
24 |
23
|
syldbl2 |
⊢ ( ( 𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ) → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 ) |