Step |
Hyp |
Ref |
Expression |
1 |
|
0ss |
⊢ ∅ ⊆ 𝐵 |
2 |
|
sspsstr |
⊢ ( ( ∅ ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴 ) → ∅ ⊊ 𝐴 ) |
3 |
1 2
|
mpan |
⊢ ( 𝐵 ⊊ 𝐴 → ∅ ⊊ 𝐴 ) |
4 |
|
0pss |
⊢ ( ∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅ ) |
5 |
|
df-ne |
⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ ) |
6 |
4 5
|
bitri |
⊢ ( ∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅ ) |
7 |
3 6
|
sylib |
⊢ ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 = ∅ ) |
8 |
|
nn0suc |
⊢ ( 𝐴 ∈ ω → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 ) ) |
9 |
8
|
orcanai |
⊢ ( ( 𝐴 ∈ ω ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 ) |
10 |
7 9
|
sylan2 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 ) |
11 |
|
pssnel |
⊢ ( 𝐵 ⊊ suc 𝑥 → ∃ 𝑦 ( 𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵 ) ) |
12 |
|
pssss |
⊢ ( 𝐵 ⊊ suc 𝑥 → 𝐵 ⊆ suc 𝑥 ) |
13 |
|
ssdif |
⊢ ( 𝐵 ⊆ suc 𝑥 → ( 𝐵 ∖ { 𝑦 } ) ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) |
14 |
|
disjsn |
⊢ ( ( 𝐵 ∩ { 𝑦 } ) = ∅ ↔ ¬ 𝑦 ∈ 𝐵 ) |
15 |
|
disj3 |
⊢ ( ( 𝐵 ∩ { 𝑦 } ) = ∅ ↔ 𝐵 = ( 𝐵 ∖ { 𝑦 } ) ) |
16 |
14 15
|
bitr3i |
⊢ ( ¬ 𝑦 ∈ 𝐵 ↔ 𝐵 = ( 𝐵 ∖ { 𝑦 } ) ) |
17 |
|
sseq1 |
⊢ ( 𝐵 = ( 𝐵 ∖ { 𝑦 } ) → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( 𝐵 ∖ { 𝑦 } ) ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
18 |
16 17
|
sylbi |
⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( 𝐵 ∖ { 𝑦 } ) ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
19 |
13 18
|
imbitrrid |
⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊆ suc 𝑥 → 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
20 |
12 19
|
syl5 |
⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊊ suc 𝑥 → 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
21 |
|
peano2 |
⊢ ( 𝑥 ∈ ω → suc 𝑥 ∈ ω ) |
22 |
|
nnfi |
⊢ ( suc 𝑥 ∈ ω → suc 𝑥 ∈ Fin ) |
23 |
|
diffi |
⊢ ( suc 𝑥 ∈ Fin → ( suc 𝑥 ∖ { 𝑦 } ) ∈ Fin ) |
24 |
|
ssdomfi |
⊢ ( ( suc 𝑥 ∖ { 𝑦 } ) ∈ Fin → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
25 |
21 22 23 24
|
4syl |
⊢ ( 𝑥 ∈ ω → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
26 |
20 25
|
sylan9 |
⊢ ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ ω ) → ( 𝐵 ⊊ suc 𝑥 → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) ) |
27 |
26
|
3impia |
⊢ ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ ω ∧ 𝐵 ⊊ suc 𝑥 ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) |
28 |
27
|
3com23 |
⊢ ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥 ∧ 𝑥 ∈ ω ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) |
29 |
28
|
3expa |
⊢ ( ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥 ) ∧ 𝑥 ∈ ω ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) |
30 |
29
|
adantrr |
⊢ ( ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) |
31 |
|
nnfi |
⊢ ( 𝑥 ∈ ω → 𝑥 ∈ Fin ) |
32 |
31
|
ad2antrl |
⊢ ( ( 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝑥 ∈ Fin ) |
33 |
|
simpl |
⊢ ( ( 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) |
34 |
|
simpr |
⊢ ( ( 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) |
35 |
|
phplem1 |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) → 𝑥 ≈ ( suc 𝑥 ∖ { 𝑦 } ) ) |
36 |
|
ensymfib |
⊢ ( 𝑥 ∈ Fin → ( 𝑥 ≈ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 ) ) |
37 |
31 36
|
syl |
⊢ ( 𝑥 ∈ ω → ( 𝑥 ≈ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) → ( 𝑥 ≈ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 ) ) |
39 |
35 38
|
mpbid |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) → ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 ) |
40 |
|
endom |
⊢ ( ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 → ( suc 𝑥 ∖ { 𝑦 } ) ≼ 𝑥 ) |
41 |
39 40
|
syl |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) → ( suc 𝑥 ∖ { 𝑦 } ) ≼ 𝑥 ) |
42 |
|
domtrfir |
⊢ ( ( 𝑥 ∈ Fin ∧ 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( suc 𝑥 ∖ { 𝑦 } ) ≼ 𝑥 ) → 𝐵 ≼ 𝑥 ) |
43 |
41 42
|
syl3an3 |
⊢ ( ( 𝑥 ∈ Fin ∧ 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝐵 ≼ 𝑥 ) |
44 |
32 33 34 43
|
syl3anc |
⊢ ( ( 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝐵 ≼ 𝑥 ) |
45 |
30 44
|
sylancom |
⊢ ( ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝐵 ≼ 𝑥 ) |
46 |
45
|
exp43 |
⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → ( 𝑦 ∈ suc 𝑥 → 𝐵 ≼ 𝑥 ) ) ) ) |
47 |
46
|
com4r |
⊢ ( 𝑦 ∈ suc 𝑥 → ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) ) ) |
48 |
47
|
imp |
⊢ ( ( 𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵 ) → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) ) |
49 |
48
|
exlimiv |
⊢ ( ∃ 𝑦 ( 𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵 ) → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) ) |
50 |
11 49
|
mpcom |
⊢ ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) |
51 |
|
simp1 |
⊢ ( ( 𝑥 ∈ ω ∧ suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → 𝑥 ∈ ω ) |
52 |
|
endom |
⊢ ( suc 𝑥 ≈ 𝐵 → suc 𝑥 ≼ 𝐵 ) |
53 |
|
domtrfir |
⊢ ( ( 𝑥 ∈ Fin ∧ suc 𝑥 ≼ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → suc 𝑥 ≼ 𝑥 ) |
54 |
52 53
|
syl3an2 |
⊢ ( ( 𝑥 ∈ Fin ∧ suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → suc 𝑥 ≼ 𝑥 ) |
55 |
31 54
|
syl3an1 |
⊢ ( ( 𝑥 ∈ ω ∧ suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → suc 𝑥 ≼ 𝑥 ) |
56 |
|
sssucid |
⊢ 𝑥 ⊆ suc 𝑥 |
57 |
|
ssdomfi |
⊢ ( suc 𝑥 ∈ Fin → ( 𝑥 ⊆ suc 𝑥 → 𝑥 ≼ suc 𝑥 ) ) |
58 |
22 56 57
|
mpisyl |
⊢ ( suc 𝑥 ∈ ω → 𝑥 ≼ suc 𝑥 ) |
59 |
21 58
|
syl |
⊢ ( 𝑥 ∈ ω → 𝑥 ≼ suc 𝑥 ) |
60 |
59
|
adantr |
⊢ ( ( 𝑥 ∈ ω ∧ suc 𝑥 ≼ 𝑥 ) → 𝑥 ≼ suc 𝑥 ) |
61 |
|
sbthfi |
⊢ ( ( 𝑥 ∈ Fin ∧ suc 𝑥 ≼ 𝑥 ∧ 𝑥 ≼ suc 𝑥 ) → suc 𝑥 ≈ 𝑥 ) |
62 |
31 61
|
syl3an1 |
⊢ ( ( 𝑥 ∈ ω ∧ suc 𝑥 ≼ 𝑥 ∧ 𝑥 ≼ suc 𝑥 ) → suc 𝑥 ≈ 𝑥 ) |
63 |
60 62
|
mpd3an3 |
⊢ ( ( 𝑥 ∈ ω ∧ suc 𝑥 ≼ 𝑥 ) → suc 𝑥 ≈ 𝑥 ) |
64 |
51 55 63
|
syl2anc |
⊢ ( ( 𝑥 ∈ ω ∧ suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → suc 𝑥 ≈ 𝑥 ) |
65 |
64
|
3com23 |
⊢ ( ( 𝑥 ∈ ω ∧ 𝐵 ≼ 𝑥 ∧ suc 𝑥 ≈ 𝐵 ) → suc 𝑥 ≈ 𝑥 ) |
66 |
65
|
3expia |
⊢ ( ( 𝑥 ∈ ω ∧ 𝐵 ≼ 𝑥 ) → ( suc 𝑥 ≈ 𝐵 → suc 𝑥 ≈ 𝑥 ) ) |
67 |
|
peano2b |
⊢ ( 𝑥 ∈ ω ↔ suc 𝑥 ∈ ω ) |
68 |
|
nnord |
⊢ ( suc 𝑥 ∈ ω → Ord suc 𝑥 ) |
69 |
67 68
|
sylbi |
⊢ ( 𝑥 ∈ ω → Ord suc 𝑥 ) |
70 |
|
vex |
⊢ 𝑥 ∈ V |
71 |
70
|
sucid |
⊢ 𝑥 ∈ suc 𝑥 |
72 |
|
nordeq |
⊢ ( ( Ord suc 𝑥 ∧ 𝑥 ∈ suc 𝑥 ) → suc 𝑥 ≠ 𝑥 ) |
73 |
69 71 72
|
sylancl |
⊢ ( 𝑥 ∈ ω → suc 𝑥 ≠ 𝑥 ) |
74 |
|
nneneq |
⊢ ( ( suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω ) → ( suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥 ) ) |
75 |
67 74
|
sylanb |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑥 ∈ ω ) → ( suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥 ) ) |
76 |
75
|
anidms |
⊢ ( 𝑥 ∈ ω → ( suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥 ) ) |
77 |
76
|
necon3bbid |
⊢ ( 𝑥 ∈ ω → ( ¬ suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 ≠ 𝑥 ) ) |
78 |
73 77
|
mpbird |
⊢ ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝑥 ) |
79 |
66 78
|
nsyli |
⊢ ( ( 𝑥 ∈ ω ∧ 𝐵 ≼ 𝑥 ) → ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵 ) ) |
80 |
79
|
expcom |
⊢ ( 𝐵 ≼ 𝑥 → ( 𝑥 ∈ ω → ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵 ) ) ) |
81 |
80
|
pm2.43d |
⊢ ( 𝐵 ≼ 𝑥 → ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵 ) ) |
82 |
50 81
|
syli |
⊢ ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵 ) ) |
83 |
82
|
com12 |
⊢ ( 𝑥 ∈ ω → ( 𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵 ) ) |
84 |
|
psseq2 |
⊢ ( 𝐴 = suc 𝑥 → ( 𝐵 ⊊ 𝐴 ↔ 𝐵 ⊊ suc 𝑥 ) ) |
85 |
|
breq1 |
⊢ ( 𝐴 = suc 𝑥 → ( 𝐴 ≈ 𝐵 ↔ suc 𝑥 ≈ 𝐵 ) ) |
86 |
85
|
notbid |
⊢ ( 𝐴 = suc 𝑥 → ( ¬ 𝐴 ≈ 𝐵 ↔ ¬ suc 𝑥 ≈ 𝐵 ) ) |
87 |
84 86
|
imbi12d |
⊢ ( 𝐴 = suc 𝑥 → ( ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ↔ ( 𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵 ) ) ) |
88 |
83 87
|
syl5ibrcom |
⊢ ( 𝑥 ∈ ω → ( 𝐴 = suc 𝑥 → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ) ) |
89 |
88
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ) |
90 |
10 89
|
syl |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ) |
91 |
90
|
syldbl2 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 ) |