Theorem dfon3 35360

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

Assertion

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

Proof of Theorem dfon3

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfon2 35260	. 2 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
2		eqabcb 2867	. . 3 ⊢ ({𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ∀𝑥(∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))))
3		vex 3470	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elrn 5884	. . . . . 6 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5		brin 5191	. . . . . . . . . . 11 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥))
6	3	brsset 35357	. . . . . . . . . . . 12 ⊢ (𝑦 SSet 𝑥 ↔ 𝑦 ⊆ 𝑥)
7		brxp 5716	. . . . . . . . . . . . . 14 ⊢ (𝑦( Trans × V)𝑥 ↔ (𝑦 ∈ Trans ∧ 𝑥 ∈ V))
8	3, 7	mpbiran2 707	. . . . . . . . . . . . 13 ⊢ (𝑦( Trans × V)𝑥 ↔ 𝑦 ∈ Trans )
9		vex 3470	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
10	9	eltrans 35359	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ Trans ↔ Tr 𝑦)
11	8, 10	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
12	6, 11	anbi12i 626	. . . . . . . . . . 11 ⊢ ((𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥) ↔ (𝑦 ⊆ 𝑥 ∧ Tr 𝑦))
13	5, 12	bitri 275	. . . . . . . . . 10 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ Tr 𝑦))
14		ioran 980	. . . . . . . . . . 11 ⊢ (¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
15		brun 5190	. . . . . . . . . . . 12 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥 ∨ 𝑦 E 𝑥))
16	3	ideq 5843	. . . . . . . . . . . . 13 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
17		epel 5574	. . . . . . . . . . . . 13 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
18	16, 17	orbi12i 911	. . . . . . . . . . . 12 ⊢ ((𝑦 I 𝑥 ∨ 𝑦 E 𝑥) ↔ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥))
19	15, 18	bitri 275	. . . . . . . . . . 11 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥))
20	14, 19	xchnxbir 333	. . . . . . . . . 10 ⊢ (¬ 𝑦( I ∪ E )𝑥 ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
21	13, 20	anbi12i 626	. . . . . . . . 9 ⊢ ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
22		brdif 5192	. . . . . . . . 9 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23		dfpss2 4078	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥))
24	23	anbi1i 623	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25		an32 643	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
26	24, 25	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
27	26	anbi1i 623	. . . . . . . . . 10 ⊢ (((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ (((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦 ∈ 𝑥))
28		anass 468	. . . . . . . . . 10 ⊢ ((((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
29	27, 28	bitri 275	. . . . . . . . 9 ⊢ (((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
30	21, 22, 29	3bitr4i 303	. . . . . . . 8 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
31	30	exbii 1842	. . . . . . 7 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
32		exanali 1854	. . . . . . 7 ⊢ (∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
33	31, 32	bitri 275	. . . . . 6 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
34	4, 33	bitri 275	. . . . 5 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
35	34	con2bii 357	. . . 4 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
36		eldif 3951	. . . . 5 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
37	3, 36	mpbiran 706	. . . 4 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
38	35, 37	bitr4i 278	. . 3 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
39	2, 38	mpgbir 1793	. 2 ⊢ {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
40	1, 39	eqtri 2752	1 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2701  Vcvv 3466   ∖ cdif 3938   ∪ cun 3939   ∩ cin 3940   ⊆ wss 3941   ⊊ wpss 3942   class class class wbr 5139  Tr wtr 5256   I cid 5564   E cep 5570   × cxp 5665  ran crn 5668  Oncon0 6355   SSet csset 35300   Trans ctrans 35301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-1st 7969  df-2nd 7970  df-txp 35322  df-sset 35324  df-trans 35325

This theorem is referenced by:  dfon4  35361