Metamath Proof Explorer


Theorem dfon3

Description: A quantifier-free definition of On . (Contributed by Scott Fenton, 5-Apr-2012)

Ref Expression
Assertion dfon3 On = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) )

Proof

Step Hyp Ref Expression
1 dfon2 On = { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) }
2 abeq1 ( { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) } = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ↔ ∀ 𝑥 ( ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) ↔ 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ) )
3 vex 𝑥 ∈ V
4 3 elrn ( 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ↔ ∃ 𝑦 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 )
5 brin ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ↔ ( 𝑦 SSet 𝑥𝑦 ( Trans × V ) 𝑥 ) )
6 3 brsset ( 𝑦 SSet 𝑥𝑦𝑥 )
7 brxp ( 𝑦 ( Trans × V ) 𝑥 ↔ ( 𝑦 Trans 𝑥 ∈ V ) )
8 3 7 mpbiran2 ( 𝑦 ( Trans × V ) 𝑥𝑦 Trans )
9 vex 𝑦 ∈ V
10 9 eltrans ( 𝑦 Trans ↔ Tr 𝑦 )
11 8 10 bitri ( 𝑦 ( Trans × V ) 𝑥 ↔ Tr 𝑦 )
12 6 11 anbi12i ( ( 𝑦 SSet 𝑥𝑦 ( Trans × V ) 𝑥 ) ↔ ( 𝑦𝑥 ∧ Tr 𝑦 ) )
13 5 12 bitri ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ↔ ( 𝑦𝑥 ∧ Tr 𝑦 ) )
14 ioran ( ¬ ( 𝑦 = 𝑥𝑦𝑥 ) ↔ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥 ) )
15 brun ( 𝑦 ( I ∪ E ) 𝑥 ↔ ( 𝑦 I 𝑥𝑦 E 𝑥 ) )
16 3 ideq ( 𝑦 I 𝑥𝑦 = 𝑥 )
17 epel ( 𝑦 E 𝑥𝑦𝑥 )
18 16 17 orbi12i ( ( 𝑦 I 𝑥𝑦 E 𝑥 ) ↔ ( 𝑦 = 𝑥𝑦𝑥 ) )
19 15 18 bitri ( 𝑦 ( I ∪ E ) 𝑥 ↔ ( 𝑦 = 𝑥𝑦𝑥 ) )
20 14 19 xchnxbir ( ¬ 𝑦 ( I ∪ E ) 𝑥 ↔ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥 ) )
21 13 20 anbi12i ( ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ∧ ¬ 𝑦 ( I ∪ E ) 𝑥 ) ↔ ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥 ) ) )
22 brdif ( 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ( 𝑦 ( SSet ∩ ( Trans × V ) ) 𝑥 ∧ ¬ 𝑦 ( I ∪ E ) 𝑥 ) )
23 dfpss2 ( 𝑦𝑥 ↔ ( 𝑦𝑥 ∧ ¬ 𝑦 = 𝑥 ) )
24 23 anbi1i ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ↔ ( ( 𝑦𝑥 ∧ ¬ 𝑦 = 𝑥 ) ∧ Tr 𝑦 ) )
25 an32 ( ( ( 𝑦𝑥 ∧ ¬ 𝑦 = 𝑥 ) ∧ Tr 𝑦 ) ↔ ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) )
26 24 25 bitri ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ↔ ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) )
27 26 anbi1i ( ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦𝑥 ) ↔ ( ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) ∧ ¬ 𝑦𝑥 ) )
28 anass ( ( ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦 = 𝑥 ) ∧ ¬ 𝑦𝑥 ) ↔ ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥 ) ) )
29 27 28 bitri ( ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦𝑥 ) ↔ ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ( ¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥 ) ) )
30 21 22 29 3bitr4i ( 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦𝑥 ) )
31 30 exbii ( ∃ 𝑦 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ∃ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦𝑥 ) )
32 exanali ( ∃ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) ∧ ¬ 𝑦𝑥 ) ↔ ¬ ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) )
33 31 32 bitri ( ∃ 𝑦 𝑦 ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) 𝑥 ↔ ¬ ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) )
34 4 33 bitri ( 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ↔ ¬ ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) )
35 34 con2bii ( ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) ↔ ¬ 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) )
36 eldif ( 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ↔ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) )
37 3 36 mpbiran ( 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) ↔ ¬ 𝑥 ∈ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) )
38 35 37 bitr4i ( ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) ↔ 𝑥 ∈ ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) ) )
39 2 38 mpgbir { 𝑥 ∣ ∀ 𝑦 ( ( 𝑦𝑥 ∧ Tr 𝑦 ) → 𝑦𝑥 ) } = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) )
40 1 39 eqtri On = ( V ∖ ran ( ( SSet ∩ ( Trans × V ) ) ∖ ( I ∪ E ) ) )