Theorem dfon3 34477

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 34367	. 2 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
2		eqabc 2879	. . 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 3449	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elrn 5849	. . . . . 6 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5		brin 5157	. . . . . . . . . . 11 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥))
6	3	brsset 34474	. . . . . . . . . . . 12 ⊢ (𝑦 SSet 𝑥 ↔ 𝑦 ⊆ 𝑥)
7		brxp 5681	. . . . . . . . . . . . . 14 ⊢ (𝑦( Trans × V)𝑥 ↔ (𝑦 ∈ Trans ∧ 𝑥 ∈ V))
8	3, 7	mpbiran2 708	. . . . . . . . . . . . 13 ⊢ (𝑦( Trans × V)𝑥 ↔ 𝑦 ∈ Trans )
9		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
10	9	eltrans 34476	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ Trans ↔ Tr 𝑦)
11	8, 10	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
12	6, 11	anbi12i 627	. . . . . . . . . . 11 ⊢ ((𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥) ↔ (𝑦 ⊆ 𝑥 ∧ Tr 𝑦))
13	5, 12	bitri 274	. . . . . . . . . 10 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ Tr 𝑦))
14		ioran 982	. . . . . . . . . . 11 ⊢ (¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
15		brun 5156	. . . . . . . . . . . 12 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥 ∨ 𝑦 E 𝑥))
16	3	ideq 5808	. . . . . . . . . . . . 13 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
17		epel 5540	. . . . . . . . . . . . 13 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
18	16, 17	orbi12i 913	. . . . . . . . . . . 12 ⊢ ((𝑦 I 𝑥 ∨ 𝑦 E 𝑥) ↔ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥))
19	15, 18	bitri 274	. . . . . . . . . . 11 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥))
20	14, 19	xchnxbir 332	. . . . . . . . . 10 ⊢ (¬ 𝑦( I ∪ E )𝑥 ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
21	13, 20	anbi12i 627	. . . . . . . . 9 ⊢ ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
22		brdif 5158	. . . . . . . . 9 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23		dfpss2 4045	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥))
24	23	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25		an32 644	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
26	24, 25	bitri 274	. . . . . . . . . . 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 469	. . . . . . . . . 10 ⊢ ((((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
29	27, 28	bitri 274	. . . . . . . . 9 ⊢ (((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
30	21, 22, 29	3bitr4i 302	. . . . . . . 8 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
31	30	exbii 1850	. . . . . . 7 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
32		exanali 1862	. . . . . . 7 ⊢ (∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
33	31, 32	bitri 274	. . . . . 6 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
34	4, 33	bitri 274	. . . . 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 3920	. . . . 5 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
37	3, 36	mpbiran 707	. . . 4 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
38	35, 37	bitr4i 277	. . 3 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
39	2, 38	mpgbir 1801	. 2 ⊢ {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
40	1, 39	eqtri 2764	1 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910   ⊊ wpss 3911   class class class wbr 5105  Tr wtr 5222   I cid 5530   E cep 5536   × cxp 5631  ran crn 5634  Oncon0 6317   SSet csset 34417   Trans ctrans 34418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-1st 7921  df-2nd 7922  df-txp 34439  df-sset 34441  df-trans 34442

This theorem is referenced by:  dfon4  34478