Step |
Hyp |
Ref |
Expression |
1 |
|
mreexexlem2d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
2 |
|
mreexexlem2d.2 |
⊢ 𝑁 = ( mrCls ‘ 𝐴 ) |
3 |
|
mreexexlem2d.3 |
⊢ 𝐼 = ( mrInd ‘ 𝐴 ) |
4 |
|
mreexexlem2d.4 |
⊢ ( 𝜑 → ∀ 𝑠 ∈ 𝒫 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) ) |
5 |
|
mreexexlem2d.5 |
⊢ ( 𝜑 → 𝐹 ⊆ ( 𝑋 ∖ 𝐻 ) ) |
6 |
|
mreexexlem2d.6 |
⊢ ( 𝜑 → 𝐺 ⊆ ( 𝑋 ∖ 𝐻 ) ) |
7 |
|
mreexexlem2d.7 |
⊢ ( 𝜑 → 𝐹 ⊆ ( 𝑁 ‘ ( 𝐺 ∪ 𝐻 ) ) ) |
8 |
|
mreexexlem2d.8 |
⊢ ( 𝜑 → ( 𝐹 ∪ 𝐻 ) ∈ 𝐼 ) |
9 |
|
mreexexlem2d.9 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐹 ) |
10 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝐹 ⊆ ( 𝑁 ‘ ( 𝐺 ∪ 𝐻 ) ) ) |
11 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
13 |
|
ssun2 |
⊢ 𝐻 ⊆ ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) |
14 |
|
difundir |
⊢ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) = ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∖ { 𝑌 } ) ) |
15 |
|
incom |
⊢ ( 𝐹 ∩ 𝐻 ) = ( 𝐻 ∩ 𝐹 ) |
16 |
|
ssdifin0 |
⊢ ( 𝐹 ⊆ ( 𝑋 ∖ 𝐻 ) → ( 𝐹 ∩ 𝐻 ) = ∅ ) |
17 |
5 16
|
syl |
⊢ ( 𝜑 → ( 𝐹 ∩ 𝐻 ) = ∅ ) |
18 |
15 17
|
eqtr3id |
⊢ ( 𝜑 → ( 𝐻 ∩ 𝐹 ) = ∅ ) |
19 |
|
minel |
⊢ ( ( 𝑌 ∈ 𝐹 ∧ ( 𝐻 ∩ 𝐹 ) = ∅ ) → ¬ 𝑌 ∈ 𝐻 ) |
20 |
9 18 19
|
syl2anc |
⊢ ( 𝜑 → ¬ 𝑌 ∈ 𝐻 ) |
21 |
|
difsnb |
⊢ ( ¬ 𝑌 ∈ 𝐻 ↔ ( 𝐻 ∖ { 𝑌 } ) = 𝐻 ) |
22 |
20 21
|
sylib |
⊢ ( 𝜑 → ( 𝐻 ∖ { 𝑌 } ) = 𝐻 ) |
23 |
22
|
uneq2d |
⊢ ( 𝜑 → ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∖ { 𝑌 } ) ) = ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ) |
24 |
14 23
|
eqtrid |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) = ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ) |
25 |
13 24
|
sseqtrrid |
⊢ ( 𝜑 → 𝐻 ⊆ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) |
26 |
3 1 8
|
mrissd |
⊢ ( 𝜑 → ( 𝐹 ∪ 𝐻 ) ⊆ 𝑋 ) |
27 |
26
|
ssdifssd |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ⊆ 𝑋 ) |
28 |
1 2 27
|
mrcssidd |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
29 |
25 28
|
sstrd |
⊢ ( 𝜑 → 𝐻 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝐻 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
31 |
12 30
|
unssd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝐺 ∪ 𝐻 ) ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
32 |
11 2
|
mrcssvd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ⊆ 𝑋 ) |
33 |
11 2 31 32
|
mrcssd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝑁 ‘ ( 𝐺 ∪ 𝐻 ) ) ⊆ ( 𝑁 ‘ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) |
34 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ⊆ 𝑋 ) |
35 |
11 2 34
|
mrcidmd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝑁 ‘ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) = ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
36 |
33 35
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝑁 ‘ ( 𝐺 ∪ 𝐻 ) ) ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
37 |
10 36
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝐹 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
38 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝑌 ∈ 𝐹 ) |
39 |
37 38
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝑌 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
40 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝐹 ∪ 𝐻 ) ∈ 𝐼 ) |
41 |
|
ssun1 |
⊢ 𝐹 ⊆ ( 𝐹 ∪ 𝐻 ) |
42 |
41 38
|
sselid |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → 𝑌 ∈ ( 𝐹 ∪ 𝐻 ) ) |
43 |
2 3 11 40 42
|
ismri2dad |
⊢ ( ( 𝜑 ∧ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ¬ 𝑌 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
44 |
39 43
|
pm2.65da |
⊢ ( 𝜑 → ¬ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
45 |
|
nss |
⊢ ( ¬ 𝐺 ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ↔ ∃ 𝑔 ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) |
46 |
44 45
|
sylib |
⊢ ( 𝜑 → ∃ 𝑔 ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) |
47 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → 𝑔 ∈ 𝐺 ) |
48 |
|
ssun1 |
⊢ ( 𝐹 ∖ { 𝑌 } ) ⊆ ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) |
49 |
48 24
|
sseqtrrid |
⊢ ( 𝜑 → ( 𝐹 ∖ { 𝑌 } ) ⊆ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) |
50 |
49 28
|
sstrd |
⊢ ( 𝜑 → ( 𝐹 ∖ { 𝑌 } ) ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
51 |
50
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( 𝐹 ∖ { 𝑌 } ) ⊆ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
52 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) |
53 |
51 52
|
ssneldd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ) |
54 |
|
unass |
⊢ ( ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ∪ { 𝑔 } ) = ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) |
55 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
56 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ∀ 𝑠 ∈ 𝒫 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ ( ( 𝑁 ‘ ( 𝑠 ∪ { 𝑦 } ) ) ∖ ( 𝑁 ‘ 𝑠 ) ) 𝑦 ∈ ( 𝑁 ‘ ( 𝑠 ∪ { 𝑧 } ) ) ) |
57 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( 𝐹 ∪ 𝐻 ) ∈ 𝐼 ) |
58 |
|
difss |
⊢ ( 𝐹 ∖ { 𝑌 } ) ⊆ 𝐹 |
59 |
|
unss1 |
⊢ ( ( 𝐹 ∖ { 𝑌 } ) ⊆ 𝐹 → ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ⊆ ( 𝐹 ∪ 𝐻 ) ) |
60 |
58 59
|
mp1i |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ⊆ ( 𝐹 ∪ 𝐻 ) ) |
61 |
55 2 3 57 60
|
mrissmrid |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ∈ 𝐼 ) |
62 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → 𝐺 ⊆ ( 𝑋 ∖ 𝐻 ) ) |
63 |
62
|
difss2d |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → 𝐺 ⊆ 𝑋 ) |
64 |
63 47
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → 𝑔 ∈ 𝑋 ) |
65 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) = ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ) |
66 |
65
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) = ( 𝑁 ‘ ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ) ) |
67 |
52 66
|
neleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ) ) |
68 |
55 2 3 56 61 64 67
|
mreexmrid |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( ( ( 𝐹 ∖ { 𝑌 } ) ∪ 𝐻 ) ∪ { 𝑔 } ) ∈ 𝐼 ) |
69 |
54 68
|
eqeltrrid |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) |
70 |
47 53 69
|
jca32 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) ) → ( 𝑔 ∈ 𝐺 ∧ ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ) ) |
71 |
70
|
ex |
⊢ ( 𝜑 → ( ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ( 𝑔 ∈ 𝐺 ∧ ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ) ) ) |
72 |
71
|
eximdv |
⊢ ( 𝜑 → ( ∃ 𝑔 ( 𝑔 ∈ 𝐺 ∧ ¬ 𝑔 ∈ ( 𝑁 ‘ ( ( 𝐹 ∪ 𝐻 ) ∖ { 𝑌 } ) ) ) → ∃ 𝑔 ( 𝑔 ∈ 𝐺 ∧ ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ) ) ) |
73 |
46 72
|
mpd |
⊢ ( 𝜑 → ∃ 𝑔 ( 𝑔 ∈ 𝐺 ∧ ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ) ) |
74 |
|
df-rex |
⊢ ( ∃ 𝑔 ∈ 𝐺 ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ↔ ∃ 𝑔 ( 𝑔 ∈ 𝐺 ∧ ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ) ) |
75 |
73 74
|
sylibr |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐺 ( ¬ 𝑔 ∈ ( 𝐹 ∖ { 𝑌 } ) ∧ ( ( 𝐹 ∖ { 𝑌 } ) ∪ ( 𝐻 ∪ { 𝑔 } ) ) ∈ 𝐼 ) ) |