Theorem dfon3 34194

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 33768	. 2 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
2		abeq1 2873	. . 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 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elrn 5802	. . . . . 6 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5		brin 5126	. . . . . . . . . . 11 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥))
6	3	brsset 34191	. . . . . . . . . . . 12 ⊢ (𝑦 SSet 𝑥 ↔ 𝑦 ⊆ 𝑥)
7		brxp 5636	. . . . . . . . . . . . . 14 ⊢ (𝑦( Trans × V)𝑥 ↔ (𝑦 ∈ Trans ∧ 𝑥 ∈ V))
8	3, 7	mpbiran2 707	. . . . . . . . . . . . 13 ⊢ (𝑦( Trans × V)𝑥 ↔ 𝑦 ∈ Trans )
9		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
10	9	eltrans 34193	. . . . . . . . . . . . 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 981	. . . . . . . . . . 11 ⊢ (¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
15		brun 5125	. . . . . . . . . . . 12 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥 ∨ 𝑦 E 𝑥))
16	3	ideq 5761	. . . . . . . . . . . . 13 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
17		epel 5498	. . . . . . . . . . . . 13 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
18	16, 17	orbi12i 912	. . . . . . . . . . . 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 333	. . . . . . . . . 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 5127	. . . . . . . . 9 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23		dfpss2 4020	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥))
24	23	anbi1i 624	. . . . . . . . . . . 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 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 303	. . . . . . . 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 358	. . . 4 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
36		eldif 3897	. . . . 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 277	. . 3 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
39	2, 38	mpgbir 1802	. 2 ⊢ {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
40	1, 39	eqtri 2766	1 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887   ⊊ wpss 3888   class class class wbr 5074  Tr wtr 5191   I cid 5488   E cep 5494   × cxp 5587  ran crn 5590  Oncon0 6266   SSet csset 34134   Trans ctrans 34135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-txp 34156  df-sset 34158  df-trans 34159

This theorem is referenced by:  dfon4  34195