Step |
Hyp |
Ref |
Expression |
1 |
|
ordtval.1 |
⊢ 𝑋 = dom 𝑅 |
2 |
|
ordtval.2 |
⊢ 𝐴 = ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) |
3 |
|
ordtval.3 |
⊢ 𝐵 = ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) |
4 |
|
ordtval.4 |
⊢ 𝐶 = ran ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
5 |
|
snex |
⊢ { 𝑋 } ∈ V |
6 |
|
ssun2 |
⊢ ( 𝐴 ∪ 𝐵 ) ⊆ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) |
7 |
1 2 3
|
ordtuni |
⊢ ( 𝑅 ∈ TosetRel → 𝑋 = ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) |
8 |
|
dmexg |
⊢ ( 𝑅 ∈ TosetRel → dom 𝑅 ∈ V ) |
9 |
1 8
|
eqeltrid |
⊢ ( 𝑅 ∈ TosetRel → 𝑋 ∈ V ) |
10 |
7 9
|
eqeltrrd |
⊢ ( 𝑅 ∈ TosetRel → ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
11 |
|
uniexb |
⊢ ( ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ↔ ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
12 |
10 11
|
sylibr |
⊢ ( 𝑅 ∈ TosetRel → ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
13 |
|
ssexg |
⊢ ( ( ( 𝐴 ∪ 𝐵 ) ⊆ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∧ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) → ( 𝐴 ∪ 𝐵 ) ∈ V ) |
14 |
6 12 13
|
sylancr |
⊢ ( 𝑅 ∈ TosetRel → ( 𝐴 ∪ 𝐵 ) ∈ V ) |
15 |
|
elfiun |
⊢ ( ( { 𝑋 } ∈ V ∧ ( 𝐴 ∪ 𝐵 ) ∈ V ) → ( 𝑧 ∈ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ↔ ( 𝑧 ∈ ( fi ‘ { 𝑋 } ) ∨ 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ ∃ 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∃ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) 𝑧 = ( 𝑚 ∩ 𝑛 ) ) ) ) |
16 |
5 14 15
|
sylancr |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ↔ ( 𝑧 ∈ ( fi ‘ { 𝑋 } ) ∨ 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ ∃ 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∃ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) 𝑧 = ( 𝑚 ∩ 𝑛 ) ) ) ) |
17 |
|
fisn |
⊢ ( fi ‘ { 𝑋 } ) = { 𝑋 } |
18 |
|
ssun1 |
⊢ { 𝑋 } ⊆ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
19 |
17 18
|
eqsstri |
⊢ ( fi ‘ { 𝑋 } ) ⊆ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
20 |
19
|
sseli |
⊢ ( 𝑧 ∈ ( fi ‘ { 𝑋 } ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
21 |
20
|
a1i |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ { 𝑋 } ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
22 |
1 2 3 4
|
ordtbas2 |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
23 |
|
ssun2 |
⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ⊆ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
24 |
22 23
|
eqsstrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
25 |
24
|
sseld |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
26 |
|
fipwuni |
⊢ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ 𝒫 ∪ ( 𝐴 ∪ 𝐵 ) |
27 |
26
|
sseli |
⊢ ( 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) → 𝑛 ∈ 𝒫 ∪ ( 𝐴 ∪ 𝐵 ) ) |
28 |
27
|
elpwid |
⊢ ( 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) → 𝑛 ⊆ ∪ ( 𝐴 ∪ 𝐵 ) ) |
29 |
28
|
ad2antll |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑛 ⊆ ∪ ( 𝐴 ∪ 𝐵 ) ) |
30 |
6
|
unissi |
⊢ ∪ ( 𝐴 ∪ 𝐵 ) ⊆ ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) |
31 |
30 7
|
sseqtrrid |
⊢ ( 𝑅 ∈ TosetRel → ∪ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑋 ) |
32 |
31
|
adantr |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → ∪ ( 𝐴 ∪ 𝐵 ) ⊆ 𝑋 ) |
33 |
29 32
|
sstrd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑛 ⊆ 𝑋 ) |
34 |
|
simprl |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑚 ∈ ( fi ‘ { 𝑋 } ) ) |
35 |
34 17
|
eleqtrdi |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑚 ∈ { 𝑋 } ) |
36 |
|
elsni |
⊢ ( 𝑚 ∈ { 𝑋 } → 𝑚 = 𝑋 ) |
37 |
35 36
|
syl |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑚 = 𝑋 ) |
38 |
33 37
|
sseqtrrd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑛 ⊆ 𝑚 ) |
39 |
|
sseqin2 |
⊢ ( 𝑛 ⊆ 𝑚 ↔ ( 𝑚 ∩ 𝑛 ) = 𝑛 ) |
40 |
38 39
|
sylib |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → ( 𝑚 ∩ 𝑛 ) = 𝑛 ) |
41 |
24
|
sselda |
⊢ ( ( 𝑅 ∈ TosetRel ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) → 𝑛 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
42 |
41
|
adantrl |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → 𝑛 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
43 |
40 42
|
eqeltrd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → ( 𝑚 ∩ 𝑛 ) ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
44 |
|
eleq1 |
⊢ ( 𝑧 = ( 𝑚 ∩ 𝑛 ) → ( 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ↔ ( 𝑚 ∩ 𝑛 ) ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
45 |
43 44
|
syl5ibrcom |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∧ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) ) → ( 𝑧 = ( 𝑚 ∩ 𝑛 ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
46 |
45
|
rexlimdvva |
⊢ ( 𝑅 ∈ TosetRel → ( ∃ 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∃ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) 𝑧 = ( 𝑚 ∩ 𝑛 ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
47 |
21 25 46
|
3jaod |
⊢ ( 𝑅 ∈ TosetRel → ( ( 𝑧 ∈ ( fi ‘ { 𝑋 } ) ∨ 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ∨ ∃ 𝑚 ∈ ( fi ‘ { 𝑋 } ) ∃ 𝑛 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) 𝑧 = ( 𝑚 ∩ 𝑛 ) ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
48 |
16 47
|
sylbid |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) → 𝑧 ∈ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) ) |
49 |
48
|
ssrdv |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ⊆ ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
50 |
|
ssfii |
⊢ ( ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V → ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
51 |
12 50
|
syl |
⊢ ( 𝑅 ∈ TosetRel → ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
52 |
51
|
unssad |
⊢ ( 𝑅 ∈ TosetRel → { 𝑋 } ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
53 |
|
fiss |
⊢ ( ( ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ∧ ( 𝐴 ∪ 𝐵 ) ⊆ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) → ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
54 |
12 6 53
|
sylancl |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
55 |
22 54
|
eqsstrrd |
⊢ ( 𝑅 ∈ TosetRel → ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
56 |
52 55
|
unssd |
⊢ ( 𝑅 ∈ TosetRel → ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) ) |
57 |
49 56
|
eqssd |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) = ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
58 |
|
unass |
⊢ ( ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∪ 𝐶 ) = ( { 𝑋 } ∪ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
59 |
57 58
|
eqtr4di |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) = ( ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∪ 𝐶 ) ) |