Step |
Hyp |
Ref |
Expression |
1 |
|
breq2 |
⊢ ( 𝑛 = ∅ → ( 𝑥 ≈ 𝑛 ↔ 𝑥 ≈ ∅ ) ) |
2 |
1
|
anbi2d |
⊢ ( 𝑛 = ∅ → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅ ) ) ) |
3 |
2
|
exbidv |
⊢ ( 𝑛 = ∅ → ( ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅ ) ) ) |
4 |
|
breq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 ≈ 𝑛 ↔ 𝑥 ≈ 𝑚 ) ) |
5 |
4
|
anbi2d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ) ) ) |
6 |
5
|
exbidv |
⊢ ( 𝑛 = 𝑚 → ( ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ) ) ) |
7 |
|
sseq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴 ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴 ) ) |
9 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≈ 𝑛 ↔ 𝑦 ≈ 𝑛 ) ) |
10 |
|
breq2 |
⊢ ( 𝑛 = suc 𝑚 → ( 𝑦 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚 ) ) |
11 |
9 10
|
sylan9bbr |
⊢ ( ( 𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚 ) ) |
12 |
8 11
|
anbi12d |
⊢ ( ( 𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ↔ ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) |
13 |
12
|
cbvexdvaw |
⊢ ( 𝑛 = suc 𝑚 → ( ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ↔ ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) |
14 |
|
0ss |
⊢ ∅ ⊆ 𝐴 |
15 |
|
peano1 |
⊢ ∅ ∈ ω |
16 |
|
enrefnn |
⊢ ( ∅ ∈ ω → ∅ ≈ ∅ ) |
17 |
15 16
|
ax-mp |
⊢ ∅ ≈ ∅ |
18 |
|
0ex |
⊢ ∅ ∈ V |
19 |
|
sseq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴 ) ) |
20 |
|
breq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 ≈ ∅ ↔ ∅ ≈ ∅ ) ) |
21 |
19 20
|
anbi12d |
⊢ ( 𝑥 = ∅ → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅ ) ↔ ( ∅ ⊆ 𝐴 ∧ ∅ ≈ ∅ ) ) ) |
22 |
18 21
|
spcev |
⊢ ( ( ∅ ⊆ 𝐴 ∧ ∅ ≈ ∅ ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅ ) ) |
23 |
14 17 22
|
mp2an |
⊢ ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅ ) |
24 |
23
|
a1i |
⊢ ( ¬ 𝐴 ∈ Fin → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅ ) ) |
25 |
|
ssdif0 |
⊢ ( 𝐴 ⊆ 𝑥 ↔ ( 𝐴 ∖ 𝑥 ) = ∅ ) |
26 |
|
eqss |
⊢ ( 𝑥 = 𝐴 ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥 ) ) |
27 |
|
breq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≈ 𝑚 ↔ 𝐴 ≈ 𝑚 ) ) |
28 |
27
|
biimpa |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚 ) → 𝐴 ≈ 𝑚 ) |
29 |
|
rspe |
⊢ ( ( 𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚 ) → ∃ 𝑚 ∈ ω 𝐴 ≈ 𝑚 ) |
30 |
28 29
|
sylan2 |
⊢ ( ( 𝑚 ∈ ω ∧ ( 𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚 ) ) → ∃ 𝑚 ∈ ω 𝐴 ≈ 𝑚 ) |
31 |
|
isfi |
⊢ ( 𝐴 ∈ Fin ↔ ∃ 𝑚 ∈ ω 𝐴 ≈ 𝑚 ) |
32 |
30 31
|
sylibr |
⊢ ( ( 𝑚 ∈ ω ∧ ( 𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚 ) ) → 𝐴 ∈ Fin ) |
33 |
32
|
expcom |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚 ) → ( 𝑚 ∈ ω → 𝐴 ∈ Fin ) ) |
34 |
26 33
|
sylanbr |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥 ) ∧ 𝑥 ≈ 𝑚 ) → ( 𝑚 ∈ ω → 𝐴 ∈ Fin ) ) |
35 |
34
|
ex |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥 ) → ( 𝑥 ≈ 𝑚 → ( 𝑚 ∈ ω → 𝐴 ∈ Fin ) ) ) |
36 |
25 35
|
sylan2br |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ ( 𝐴 ∖ 𝑥 ) = ∅ ) → ( 𝑥 ≈ 𝑚 → ( 𝑚 ∈ ω → 𝐴 ∈ Fin ) ) ) |
37 |
36
|
expcom |
⊢ ( ( 𝐴 ∖ 𝑥 ) = ∅ → ( 𝑥 ⊆ 𝐴 → ( 𝑥 ≈ 𝑚 → ( 𝑚 ∈ ω → 𝐴 ∈ Fin ) ) ) ) |
38 |
37
|
3impd |
⊢ ( ( 𝐴 ∖ 𝑥 ) = ∅ → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω ) → 𝐴 ∈ Fin ) ) |
39 |
38
|
com12 |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω ) → ( ( 𝐴 ∖ 𝑥 ) = ∅ → 𝐴 ∈ Fin ) ) |
40 |
39
|
con3d |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω ) → ( ¬ 𝐴 ∈ Fin → ¬ ( 𝐴 ∖ 𝑥 ) = ∅ ) ) |
41 |
|
bren |
⊢ ( 𝑥 ≈ 𝑚 ↔ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) |
42 |
|
neq0 |
⊢ ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ ↔ ∃ 𝑧 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ) |
43 |
|
eldifi |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) → 𝑧 ∈ 𝐴 ) |
44 |
43
|
snssd |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) → { 𝑧 } ⊆ 𝐴 ) |
45 |
|
unss |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ { 𝑧 } ⊆ 𝐴 ) ↔ ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
46 |
45
|
biimpi |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ { 𝑧 } ⊆ 𝐴 ) → ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
47 |
44 46
|
sylan2 |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ) → ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
48 |
47
|
ad2ant2r |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) ∧ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) ) → ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
49 |
|
vex |
⊢ 𝑧 ∈ V |
50 |
|
vex |
⊢ 𝑚 ∈ V |
51 |
49 50
|
f1osn |
⊢ { 〈 𝑧 , 𝑚 〉 } : { 𝑧 } –1-1-onto→ { 𝑚 } |
52 |
51
|
jctr |
⊢ ( 𝑓 : 𝑥 –1-1-onto→ 𝑚 → ( 𝑓 : 𝑥 –1-1-onto→ 𝑚 ∧ { 〈 𝑧 , 𝑚 〉 } : { 𝑧 } –1-1-onto→ { 𝑚 } ) ) |
53 |
|
eldifn |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) → ¬ 𝑧 ∈ 𝑥 ) |
54 |
|
disjsn |
⊢ ( ( 𝑥 ∩ { 𝑧 } ) = ∅ ↔ ¬ 𝑧 ∈ 𝑥 ) |
55 |
53 54
|
sylibr |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) → ( 𝑥 ∩ { 𝑧 } ) = ∅ ) |
56 |
|
nnord |
⊢ ( 𝑚 ∈ ω → Ord 𝑚 ) |
57 |
|
orddisj |
⊢ ( Ord 𝑚 → ( 𝑚 ∩ { 𝑚 } ) = ∅ ) |
58 |
56 57
|
syl |
⊢ ( 𝑚 ∈ ω → ( 𝑚 ∩ { 𝑚 } ) = ∅ ) |
59 |
55 58
|
anim12i |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) → ( ( 𝑥 ∩ { 𝑧 } ) = ∅ ∧ ( 𝑚 ∩ { 𝑚 } ) = ∅ ) ) |
60 |
|
f1oun |
⊢ ( ( ( 𝑓 : 𝑥 –1-1-onto→ 𝑚 ∧ { 〈 𝑧 , 𝑚 〉 } : { 𝑧 } –1-1-onto→ { 𝑚 } ) ∧ ( ( 𝑥 ∩ { 𝑧 } ) = ∅ ∧ ( 𝑚 ∩ { 𝑚 } ) = ∅ ) ) → ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ ( 𝑚 ∪ { 𝑚 } ) ) |
61 |
52 59 60
|
syl2an |
⊢ ( ( 𝑓 : 𝑥 –1-1-onto→ 𝑚 ∧ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) ) → ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ ( 𝑚 ∪ { 𝑚 } ) ) |
62 |
|
df-suc |
⊢ suc 𝑚 = ( 𝑚 ∪ { 𝑚 } ) |
63 |
|
f1oeq3 |
⊢ ( suc 𝑚 = ( 𝑚 ∪ { 𝑚 } ) → ( ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 ↔ ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ ( 𝑚 ∪ { 𝑚 } ) ) ) |
64 |
62 63
|
ax-mp |
⊢ ( ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 ↔ ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ ( 𝑚 ∪ { 𝑚 } ) ) |
65 |
|
vex |
⊢ 𝑓 ∈ V |
66 |
|
snex |
⊢ { 〈 𝑧 , 𝑚 〉 } ∈ V |
67 |
65 66
|
unex |
⊢ ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) ∈ V |
68 |
|
f1oeq1 |
⊢ ( 𝑔 = ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) → ( 𝑔 : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 ↔ ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 ) ) |
69 |
67 68
|
spcev |
⊢ ( ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 → ∃ 𝑔 𝑔 : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 ) |
70 |
|
bren |
⊢ ( ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ↔ ∃ 𝑔 𝑔 : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 ) |
71 |
69 70
|
sylibr |
⊢ ( ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ suc 𝑚 → ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) |
72 |
64 71
|
sylbir |
⊢ ( ( 𝑓 ∪ { 〈 𝑧 , 𝑚 〉 } ) : ( 𝑥 ∪ { 𝑧 } ) –1-1-onto→ ( 𝑚 ∪ { 𝑚 } ) → ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) |
73 |
61 72
|
syl |
⊢ ( ( 𝑓 : 𝑥 –1-1-onto→ 𝑚 ∧ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) ) → ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) |
74 |
73
|
adantll |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) ∧ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) ) → ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) |
75 |
|
vex |
⊢ 𝑥 ∈ V |
76 |
|
snex |
⊢ { 𝑧 } ∈ V |
77 |
75 76
|
unex |
⊢ ( 𝑥 ∪ { 𝑧 } ) ∈ V |
78 |
|
sseq1 |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ( 𝑦 ⊆ 𝐴 ↔ ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ) ) |
79 |
|
breq1 |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ( 𝑦 ≈ suc 𝑚 ↔ ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) ) |
80 |
78 79
|
anbi12d |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ( ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ↔ ( ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) ) ) |
81 |
77 80
|
spcev |
⊢ ( ( ( 𝑥 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ( 𝑥 ∪ { 𝑧 } ) ≈ suc 𝑚 ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) |
82 |
48 74 81
|
syl2anc |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) ∧ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) |
83 |
82
|
expcom |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) ∧ 𝑚 ∈ ω ) → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) |
84 |
83
|
ex |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) → ( 𝑚 ∈ ω → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
85 |
84
|
exlimiv |
⊢ ( ∃ 𝑧 𝑧 ∈ ( 𝐴 ∖ 𝑥 ) → ( 𝑚 ∈ ω → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
86 |
42 85
|
sylbi |
⊢ ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ → ( 𝑚 ∈ ω → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
87 |
86
|
com13 |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑓 : 𝑥 –1-1-onto→ 𝑚 ) → ( 𝑚 ∈ ω → ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
88 |
87
|
expcom |
⊢ ( 𝑓 : 𝑥 –1-1-onto→ 𝑚 → ( 𝑥 ⊆ 𝐴 → ( 𝑚 ∈ ω → ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) ) |
89 |
88
|
exlimiv |
⊢ ( ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑚 → ( 𝑥 ⊆ 𝐴 → ( 𝑚 ∈ ω → ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) ) |
90 |
41 89
|
sylbi |
⊢ ( 𝑥 ≈ 𝑚 → ( 𝑥 ⊆ 𝐴 → ( 𝑚 ∈ ω → ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) ) |
91 |
90
|
3imp21 |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω ) → ( ¬ ( 𝐴 ∖ 𝑥 ) = ∅ → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) |
92 |
40 91
|
syld |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω ) → ( ¬ 𝐴 ∈ Fin → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) |
93 |
92
|
3expia |
⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ) → ( 𝑚 ∈ ω → ( ¬ 𝐴 ∈ Fin → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
94 |
93
|
exlimiv |
⊢ ( ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ) → ( 𝑚 ∈ ω → ( ¬ 𝐴 ∈ Fin → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
95 |
94
|
com3l |
⊢ ( 𝑚 ∈ ω → ( ¬ 𝐴 ∈ Fin → ( ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ) → ∃ 𝑦 ( 𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚 ) ) ) ) |
96 |
3 6 13 24 95
|
finds2 |
⊢ ( 𝑛 ∈ ω → ( ¬ 𝐴 ∈ Fin → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ) ) |
97 |
96
|
com12 |
⊢ ( ¬ 𝐴 ∈ Fin → ( 𝑛 ∈ ω → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ) ) |
98 |
97
|
ralrimiv |
⊢ ( ¬ 𝐴 ∈ Fin → ∀ 𝑛 ∈ ω ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛 ) ) |