Step |
Hyp |
Ref |
Expression |
1 |
|
reldom |
⊢ Rel ≼ |
2 |
1
|
brrelex2i |
⊢ ( 𝐴 ≼ 𝐵 → 𝐵 ∈ V ) |
3 |
|
domeng |
⊢ ( 𝐵 ∈ V → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐴 ≼ 𝐵 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ) ) |
5 |
4
|
ibi |
⊢ ( 𝐴 ≼ 𝐵 → ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ) |
6 |
1
|
brrelex1i |
⊢ ( 𝐶 ≼ 𝐷 → 𝐶 ∈ V ) |
7 |
|
difss |
⊢ ( 𝐶 ∖ 𝐴 ) ⊆ 𝐶 |
8 |
|
ssdomg |
⊢ ( 𝐶 ∈ V → ( ( 𝐶 ∖ 𝐴 ) ⊆ 𝐶 → ( 𝐶 ∖ 𝐴 ) ≼ 𝐶 ) ) |
9 |
6 7 8
|
mpisyl |
⊢ ( 𝐶 ≼ 𝐷 → ( 𝐶 ∖ 𝐴 ) ≼ 𝐶 ) |
10 |
|
domtr |
⊢ ( ( ( 𝐶 ∖ 𝐴 ) ≼ 𝐶 ∧ 𝐶 ≼ 𝐷 ) → ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 ) |
11 |
9 10
|
mpancom |
⊢ ( 𝐶 ≼ 𝐷 → ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 ) |
12 |
1
|
brrelex2i |
⊢ ( ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 → 𝐷 ∈ V ) |
13 |
|
domeng |
⊢ ( 𝐷 ∈ V → ( ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 ↔ ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 → ( ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 ↔ ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) |
15 |
14
|
ibi |
⊢ ( ( 𝐶 ∖ 𝐴 ) ≼ 𝐷 → ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) |
16 |
11 15
|
syl |
⊢ ( 𝐶 ≼ 𝐷 → ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) |
17 |
5 16
|
anim12i |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) → ( ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) |
19 |
|
exdistrv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ↔ ( ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) |
20 |
|
simprll |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → 𝐴 ≈ 𝑥 ) |
21 |
|
simprrl |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ) |
22 |
|
disjdif |
⊢ ( 𝐴 ∩ ( 𝐶 ∖ 𝐴 ) ) = ∅ |
23 |
22
|
a1i |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝐴 ∩ ( 𝐶 ∖ 𝐴 ) ) = ∅ ) |
24 |
|
ss2in |
⊢ ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐷 ) → ( 𝑥 ∩ 𝑦 ) ⊆ ( 𝐵 ∩ 𝐷 ) ) |
25 |
24
|
ad2ant2l |
⊢ ( ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) → ( 𝑥 ∩ 𝑦 ) ⊆ ( 𝐵 ∩ 𝐷 ) ) |
26 |
25
|
adantl |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝑥 ∩ 𝑦 ) ⊆ ( 𝐵 ∩ 𝐷 ) ) |
27 |
|
simplr |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝐵 ∩ 𝐷 ) = ∅ ) |
28 |
|
sseq0 |
⊢ ( ( ( 𝑥 ∩ 𝑦 ) ⊆ ( 𝐵 ∩ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( 𝑥 ∩ 𝑦 ) = ∅ ) |
29 |
26 27 28
|
syl2anc |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝑥 ∩ 𝑦 ) = ∅ ) |
30 |
|
undif2 |
⊢ ( 𝐴 ∪ ( 𝐶 ∖ 𝐴 ) ) = ( 𝐴 ∪ 𝐶 ) |
31 |
|
unen |
⊢ ( ( ( 𝐴 ≈ 𝑥 ∧ ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ) ∧ ( ( 𝐴 ∩ ( 𝐶 ∖ 𝐴 ) ) = ∅ ∧ ( 𝑥 ∩ 𝑦 ) = ∅ ) ) → ( 𝐴 ∪ ( 𝐶 ∖ 𝐴 ) ) ≈ ( 𝑥 ∪ 𝑦 ) ) |
32 |
30 31
|
eqbrtrrid |
⊢ ( ( ( 𝐴 ≈ 𝑥 ∧ ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ) ∧ ( ( 𝐴 ∩ ( 𝐶 ∖ 𝐴 ) ) = ∅ ∧ ( 𝑥 ∩ 𝑦 ) = ∅ ) ) → ( 𝐴 ∪ 𝐶 ) ≈ ( 𝑥 ∪ 𝑦 ) ) |
33 |
20 21 23 29 32
|
syl22anc |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝐴 ∪ 𝐶 ) ≈ ( 𝑥 ∪ 𝑦 ) ) |
34 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → 𝐵 ∈ V ) |
35 |
1
|
brrelex2i |
⊢ ( 𝐶 ≼ 𝐷 → 𝐷 ∈ V ) |
36 |
35
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → 𝐷 ∈ V ) |
37 |
|
unexg |
⊢ ( ( 𝐵 ∈ V ∧ 𝐷 ∈ V ) → ( 𝐵 ∪ 𝐷 ) ∈ V ) |
38 |
34 36 37
|
syl2anc |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝐵 ∪ 𝐷 ) ∈ V ) |
39 |
|
unss12 |
⊢ ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐷 ) → ( 𝑥 ∪ 𝑦 ) ⊆ ( 𝐵 ∪ 𝐷 ) ) |
40 |
39
|
ad2ant2l |
⊢ ( ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) → ( 𝑥 ∪ 𝑦 ) ⊆ ( 𝐵 ∪ 𝐷 ) ) |
41 |
40
|
adantl |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝑥 ∪ 𝑦 ) ⊆ ( 𝐵 ∪ 𝐷 ) ) |
42 |
|
ssdomg |
⊢ ( ( 𝐵 ∪ 𝐷 ) ∈ V → ( ( 𝑥 ∪ 𝑦 ) ⊆ ( 𝐵 ∪ 𝐷 ) → ( 𝑥 ∪ 𝑦 ) ≼ ( 𝐵 ∪ 𝐷 ) ) ) |
43 |
38 41 42
|
sylc |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝑥 ∪ 𝑦 ) ≼ ( 𝐵 ∪ 𝐷 ) ) |
44 |
|
endomtr |
⊢ ( ( ( 𝐴 ∪ 𝐶 ) ≈ ( 𝑥 ∪ 𝑦 ) ∧ ( 𝑥 ∪ 𝑦 ) ≼ ( 𝐵 ∪ 𝐷 ) ) → ( 𝐴 ∪ 𝐶 ) ≼ ( 𝐵 ∪ 𝐷 ) ) |
45 |
33 43 44
|
syl2anc |
⊢ ( ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ∧ ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) ) → ( 𝐴 ∪ 𝐶 ) ≼ ( 𝐵 ∪ 𝐷 ) ) |
46 |
45
|
ex |
⊢ ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) → ( 𝐴 ∪ 𝐶 ) ≼ ( 𝐵 ∪ 𝐷 ) ) ) |
47 |
46
|
exlimdvv |
⊢ ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( ∃ 𝑥 ∃ 𝑦 ( ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) → ( 𝐴 ∪ 𝐶 ) ≼ ( 𝐵 ∪ 𝐷 ) ) ) |
48 |
19 47
|
syl5bir |
⊢ ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( ( ∃ 𝑥 ( 𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ∃ 𝑦 ( ( 𝐶 ∖ 𝐴 ) ≈ 𝑦 ∧ 𝑦 ⊆ 𝐷 ) ) → ( 𝐴 ∪ 𝐶 ) ≼ ( 𝐵 ∪ 𝐷 ) ) ) |
49 |
18 48
|
mpd |
⊢ ( ( ( 𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( 𝐴 ∪ 𝐶 ) ≼ ( 𝐵 ∪ 𝐷 ) ) |