Step |
Hyp |
Ref |
Expression |
1 |
|
domunsncan.a |
⊢ 𝐴 ∈ V |
2 |
|
domunsncan.b |
⊢ 𝐵 ∈ V |
3 |
|
ssun2 |
⊢ 𝑌 ⊆ ( { 𝐵 } ∪ 𝑌 ) |
4 |
|
reldom |
⊢ Rel ≼ |
5 |
4
|
brrelex2i |
⊢ ( ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) → ( { 𝐵 } ∪ 𝑌 ) ∈ V ) |
6 |
5
|
adantl |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ) → ( { 𝐵 } ∪ 𝑌 ) ∈ V ) |
7 |
|
ssexg |
⊢ ( ( 𝑌 ⊆ ( { 𝐵 } ∪ 𝑌 ) ∧ ( { 𝐵 } ∪ 𝑌 ) ∈ V ) → 𝑌 ∈ V ) |
8 |
3 6 7
|
sylancr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ) → 𝑌 ∈ V ) |
9 |
|
brdomi |
⊢ ( ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) → ∃ 𝑓 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) |
10 |
|
vex |
⊢ 𝑓 ∈ V |
11 |
10
|
resex |
⊢ ( 𝑓 ↾ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ∈ V |
12 |
|
simprr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) |
13 |
|
difss |
⊢ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ⊆ ( { 𝐴 } ∪ 𝑋 ) |
14 |
|
f1ores |
⊢ ( ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ∧ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ⊆ ( { 𝐴 } ∪ 𝑋 ) ) → ( 𝑓 ↾ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) : ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) –1-1-onto→ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ) |
15 |
12 13 14
|
sylancl |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( 𝑓 ↾ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) : ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) –1-1-onto→ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ) |
16 |
|
f1oen3g |
⊢ ( ( ( 𝑓 ↾ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ∈ V ∧ ( 𝑓 ↾ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) : ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) –1-1-onto→ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ) → ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ≈ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ) |
17 |
11 15 16
|
sylancr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ≈ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ) |
18 |
|
df-f1 |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ↔ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) ⟶ ( { 𝐵 } ∪ 𝑌 ) ∧ Fun ◡ 𝑓 ) ) |
19 |
|
imadif |
⊢ ( Fun ◡ 𝑓 → ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) = ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
20 |
18 19
|
simplbiim |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) = ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
21 |
20
|
ad2antll |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) = ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
22 |
|
snex |
⊢ { 𝐵 } ∈ V |
23 |
|
simprl |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → 𝑌 ∈ V ) |
24 |
|
unexg |
⊢ ( ( { 𝐵 } ∈ V ∧ 𝑌 ∈ V ) → ( { 𝐵 } ∪ 𝑌 ) ∈ V ) |
25 |
22 23 24
|
sylancr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( { 𝐵 } ∪ 𝑌 ) ∈ V ) |
26 |
25
|
difexd |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ∈ V ) |
27 |
|
f1f |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → 𝑓 : ( { 𝐴 } ∪ 𝑋 ) ⟶ ( { 𝐵 } ∪ 𝑌 ) ) |
28 |
|
fimass |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) ⟶ ( { 𝐵 } ∪ 𝑌 ) → ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ⊆ ( { 𝐵 } ∪ 𝑌 ) ) |
29 |
27 28
|
syl |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ⊆ ( { 𝐵 } ∪ 𝑌 ) ) |
30 |
29
|
ad2antll |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ⊆ ( { 𝐵 } ∪ 𝑌 ) ) |
31 |
30
|
ssdifd |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ⊆ ( ( { 𝐵 } ∪ 𝑌 ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
32 |
|
f1fn |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → 𝑓 Fn ( { 𝐴 } ∪ 𝑋 ) ) |
33 |
32
|
ad2antll |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → 𝑓 Fn ( { 𝐴 } ∪ 𝑋 ) ) |
34 |
1
|
snid |
⊢ 𝐴 ∈ { 𝐴 } |
35 |
|
elun1 |
⊢ ( 𝐴 ∈ { 𝐴 } → 𝐴 ∈ ( { 𝐴 } ∪ 𝑋 ) ) |
36 |
34 35
|
ax-mp |
⊢ 𝐴 ∈ ( { 𝐴 } ∪ 𝑋 ) |
37 |
|
fnsnfv |
⊢ ( ( 𝑓 Fn ( { 𝐴 } ∪ 𝑋 ) ∧ 𝐴 ∈ ( { 𝐴 } ∪ 𝑋 ) ) → { ( 𝑓 ‘ 𝐴 ) } = ( 𝑓 “ { 𝐴 } ) ) |
38 |
33 36 37
|
sylancl |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → { ( 𝑓 ‘ 𝐴 ) } = ( 𝑓 “ { 𝐴 } ) ) |
39 |
38
|
difeq2d |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) = ( ( { 𝐵 } ∪ 𝑌 ) ∖ ( 𝑓 “ { 𝐴 } ) ) ) |
40 |
31 39
|
sseqtrrd |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ⊆ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
41 |
|
ssdomg |
⊢ ( ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ∈ V → ( ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ⊆ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) → ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) ) |
42 |
26 40 41
|
sylc |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ) |
43 |
|
ffvelrn |
⊢ ( ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) ⟶ ( { 𝐵 } ∪ 𝑌 ) ∧ 𝐴 ∈ ( { 𝐴 } ∪ 𝑋 ) ) → ( 𝑓 ‘ 𝐴 ) ∈ ( { 𝐵 } ∪ 𝑌 ) ) |
44 |
27 36 43
|
sylancl |
⊢ ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → ( 𝑓 ‘ 𝐴 ) ∈ ( { 𝐵 } ∪ 𝑌 ) ) |
45 |
44
|
ad2antll |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( 𝑓 ‘ 𝐴 ) ∈ ( { 𝐵 } ∪ 𝑌 ) ) |
46 |
2
|
snid |
⊢ 𝐵 ∈ { 𝐵 } |
47 |
|
elun1 |
⊢ ( 𝐵 ∈ { 𝐵 } → 𝐵 ∈ ( { 𝐵 } ∪ 𝑌 ) ) |
48 |
46 47
|
mp1i |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → 𝐵 ∈ ( { 𝐵 } ∪ 𝑌 ) ) |
49 |
|
difsnen |
⊢ ( ( ( { 𝐵 } ∪ 𝑌 ) ∈ V ∧ ( 𝑓 ‘ 𝐴 ) ∈ ( { 𝐵 } ∪ 𝑌 ) ∧ 𝐵 ∈ ( { 𝐵 } ∪ 𝑌 ) ) → ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
50 |
25 45 48 49
|
syl3anc |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
51 |
|
domentr |
⊢ ( ( ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ∧ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { ( 𝑓 ‘ 𝐴 ) } ) ≈ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) → ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
52 |
42 50 51
|
syl2anc |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( 𝑓 “ ( { 𝐴 } ∪ 𝑋 ) ) ∖ ( 𝑓 “ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
53 |
21 52
|
eqbrtrd |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
54 |
|
endomtr |
⊢ ( ( ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ≈ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ∧ ( 𝑓 “ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) → ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
55 |
17 53 54
|
syl2anc |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) ≼ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) ) |
56 |
|
uncom |
⊢ ( { 𝐴 } ∪ 𝑋 ) = ( 𝑋 ∪ { 𝐴 } ) |
57 |
56
|
difeq1i |
⊢ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) = ( ( 𝑋 ∪ { 𝐴 } ) ∖ { 𝐴 } ) |
58 |
|
difun2 |
⊢ ( ( 𝑋 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = ( 𝑋 ∖ { 𝐴 } ) |
59 |
57 58
|
eqtri |
⊢ ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) = ( 𝑋 ∖ { 𝐴 } ) |
60 |
|
difsn |
⊢ ( ¬ 𝐴 ∈ 𝑋 → ( 𝑋 ∖ { 𝐴 } ) = 𝑋 ) |
61 |
59 60
|
eqtrid |
⊢ ( ¬ 𝐴 ∈ 𝑋 → ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) = 𝑋 ) |
62 |
61
|
ad2antrr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐴 } ∪ 𝑋 ) ∖ { 𝐴 } ) = 𝑋 ) |
63 |
|
uncom |
⊢ ( { 𝐵 } ∪ 𝑌 ) = ( 𝑌 ∪ { 𝐵 } ) |
64 |
63
|
difeq1i |
⊢ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) = ( ( 𝑌 ∪ { 𝐵 } ) ∖ { 𝐵 } ) |
65 |
|
difun2 |
⊢ ( ( 𝑌 ∪ { 𝐵 } ) ∖ { 𝐵 } ) = ( 𝑌 ∖ { 𝐵 } ) |
66 |
64 65
|
eqtri |
⊢ ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) = ( 𝑌 ∖ { 𝐵 } ) |
67 |
|
difsn |
⊢ ( ¬ 𝐵 ∈ 𝑌 → ( 𝑌 ∖ { 𝐵 } ) = 𝑌 ) |
68 |
66 67
|
eqtrid |
⊢ ( ¬ 𝐵 ∈ 𝑌 → ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) = 𝑌 ) |
69 |
68
|
ad2antlr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → ( ( { 𝐵 } ∪ 𝑌 ) ∖ { 𝐵 } ) = 𝑌 ) |
70 |
55 62 69
|
3brtr3d |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( 𝑌 ∈ V ∧ 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) ) ) → 𝑋 ≼ 𝑌 ) |
71 |
70
|
expr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑌 ∈ V ) → ( 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → 𝑋 ≼ 𝑌 ) ) |
72 |
71
|
exlimdv |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑌 ∈ V ) → ( ∃ 𝑓 𝑓 : ( { 𝐴 } ∪ 𝑋 ) –1-1→ ( { 𝐵 } ∪ 𝑌 ) → 𝑋 ≼ 𝑌 ) ) |
73 |
9 72
|
syl5 |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑌 ∈ V ) → ( ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) → 𝑋 ≼ 𝑌 ) ) |
74 |
73
|
impancom |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ) → ( 𝑌 ∈ V → 𝑋 ≼ 𝑌 ) ) |
75 |
8 74
|
mpd |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ) → 𝑋 ≼ 𝑌 ) |
76 |
|
en2sn |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → { 𝐴 } ≈ { 𝐵 } ) |
77 |
1 2 76
|
mp2an |
⊢ { 𝐴 } ≈ { 𝐵 } |
78 |
|
endom |
⊢ ( { 𝐴 } ≈ { 𝐵 } → { 𝐴 } ≼ { 𝐵 } ) |
79 |
77 78
|
mp1i |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑋 ≼ 𝑌 ) → { 𝐴 } ≼ { 𝐵 } ) |
80 |
|
simpr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑋 ≼ 𝑌 ) → 𝑋 ≼ 𝑌 ) |
81 |
|
incom |
⊢ ( { 𝐵 } ∩ 𝑌 ) = ( 𝑌 ∩ { 𝐵 } ) |
82 |
|
disjsn |
⊢ ( ( 𝑌 ∩ { 𝐵 } ) = ∅ ↔ ¬ 𝐵 ∈ 𝑌 ) |
83 |
82
|
biimpri |
⊢ ( ¬ 𝐵 ∈ 𝑌 → ( 𝑌 ∩ { 𝐵 } ) = ∅ ) |
84 |
81 83
|
eqtrid |
⊢ ( ¬ 𝐵 ∈ 𝑌 → ( { 𝐵 } ∩ 𝑌 ) = ∅ ) |
85 |
84
|
ad2antlr |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑋 ≼ 𝑌 ) → ( { 𝐵 } ∩ 𝑌 ) = ∅ ) |
86 |
|
undom |
⊢ ( ( ( { 𝐴 } ≼ { 𝐵 } ∧ 𝑋 ≼ 𝑌 ) ∧ ( { 𝐵 } ∩ 𝑌 ) = ∅ ) → ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ) |
87 |
79 80 85 86
|
syl21anc |
⊢ ( ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) ∧ 𝑋 ≼ 𝑌 ) → ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ) |
88 |
75 87
|
impbida |
⊢ ( ( ¬ 𝐴 ∈ 𝑋 ∧ ¬ 𝐵 ∈ 𝑌 ) → ( ( { 𝐴 } ∪ 𝑋 ) ≼ ( { 𝐵 } ∪ 𝑌 ) ↔ 𝑋 ≼ 𝑌 ) ) |