Theorem dfon3 35873

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 35773	. 2 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
2		eqabcb 2880	. . 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 3481	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elrn 5906	. . . . . 6 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5		brin 5199	. . . . . . . . . . 11 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥))
6	3	brsset 35870	. . . . . . . . . . . 12 ⊢ (𝑦 SSet 𝑥 ↔ 𝑦 ⊆ 𝑥)
7		brxp 5737	. . . . . . . . . . . . . 14 ⊢ (𝑦( Trans × V)𝑥 ↔ (𝑦 ∈ Trans ∧ 𝑥 ∈ V))
8	3, 7	mpbiran2 710	. . . . . . . . . . . . 13 ⊢ (𝑦( Trans × V)𝑥 ↔ 𝑦 ∈ Trans )
9		vex 3481	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
10	9	eltrans 35872	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ Trans ↔ Tr 𝑦)
11	8, 10	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
12	6, 11	anbi12i 628	. . . . . . . . . . 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 985	. . . . . . . . . . 11 ⊢ (¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
15		brun 5198	. . . . . . . . . . . 12 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥 ∨ 𝑦 E 𝑥))
16	3	ideq 5865	. . . . . . . . . . . . 13 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
17		epel 5591	. . . . . . . . . . . . 13 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
18	16, 17	orbi12i 914	. . . . . . . . . . . 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 628	. . . . . . . . 9 ⊢ ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
22		brdif 5200	. . . . . . . . 9 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23		dfpss2 4097	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥))
24	23	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25		an32 646	. . . . . . . . . . . 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 624	. . . . . . . . . 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 1844	. . . . . . 7 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
32		exanali 1856	. . . . . . 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 3972	. . . . 5 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
37	3, 36	mpbiran 709	. . . 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 1795	. 2 ⊢ {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
40	1, 39	eqtri 2762	1 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1534   = wceq 1536  ∃wex 1775   ∈ wcel 2105  {cab 2711  Vcvv 3477   ∖ cdif 3959   ∪ cun 3960   ∩ cin 3961   ⊆ wss 3962   ⊊ wpss 3963   class class class wbr 5147  Tr wtr 5264   I cid 5581   E cep 5587   × cxp 5686  ran crn 5689  Oncon0 6385   SSet csset 35813   Trans ctrans 35814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-sset 35837  df-trans 35838

This theorem is referenced by:  dfon4  35874