Theorem dfon3 36750

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 36650	. 2 ⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
2		eqabcb 2900	. . 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 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	elrn 5876	. . . . . 6 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5		brin 5156	. . . . . . . . . . 11 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥))
6	3	brsset 36747	. . . . . . . . . . . 12 ⊢ (𝑦 SSet 𝑥 ↔ 𝑦 ⊆ 𝑥)
7		brxp 5700	. . . . . . . . . . . . . 14 ⊢ (𝑦( Trans × V)𝑥 ↔ (𝑦 ∈ Trans ∧ 𝑥 ∈ V))
8	3, 7	mpbiran2 723	. . . . . . . . . . . . 13 ⊢ (𝑦( Trans × V)𝑥 ↔ 𝑦 ∈ Trans )
9		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
10	9	eltrans 36749	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ Trans ↔ Tr 𝑦)
11	8, 10	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
12	6, 11	anbi12i 640	. . . . . . . . . . 11 ⊢ ((𝑦 SSet 𝑥 ∧ 𝑦( Trans × V)𝑥) ↔ (𝑦 ⊆ 𝑥 ∧ Tr 𝑦))
13	5, 12	bitri 278	. . . . . . . . . 10 ⊢ (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ Tr 𝑦))
14		ioran 999	. . . . . . . . . . 11 ⊢ (¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
15		brun 5155	. . . . . . . . . . . 12 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥 ∨ 𝑦 E 𝑥))
16	3	ideq 5831	. . . . . . . . . . . . 13 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
17		epel 5554	. . . . . . . . . . . . 13 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
18	16, 17	orbi12i 928	. . . . . . . . . . . 12 ⊢ ((𝑦 I 𝑥 ∨ 𝑦 E 𝑥) ↔ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥))
19	15, 18	bitri 278	. . . . . . . . . . 11 ⊢ (𝑦( I ∪ E )𝑥 ↔ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥))
20	14, 19	xchnxbir 336	. . . . . . . . . 10 ⊢ (¬ 𝑦( I ∪ E )𝑥 ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥))
21	13, 20	anbi12i 640	. . . . . . . . 9 ⊢ ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
22		brdif 5157	. . . . . . . . 9 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23		dfpss2 4014	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥))
24	23	anbi1i 636	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25		an32 659	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
26	24, 25	bitri 278	. . . . . . . . . . 11 ⊢ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
27	26	anbi1i 636	. . . . . . . . . 10 ⊢ (((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ (((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦 ∈ 𝑥))
28		anass 474	. . . . . . . . . 10 ⊢ ((((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
29	27, 28	bitri 278	. . . . . . . . 9 ⊢ (((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦 ∈ 𝑥)))
30	21, 22, 29	3bitr4i 306	. . . . . . . 8 ⊢ (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
31	30	exbii 1880	. . . . . . 7 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥))
32		exanali 1891	. . . . . . 7 ⊢ (∃𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 ∈ 𝑥) ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
33	31, 32	bitri 278	. . . . . 6 ⊢ (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
34	4, 33	bitri 278	. . . . 5 ⊢ (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ¬ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥))
35	34	con2bii 360	. . . 4 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
36		eldif 4048	. . . . 5 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
37	3, 36	mpbiran 722	. . . 4 ⊢ (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
38	35, 37	bitr4i 281	. . 3 ⊢ (∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
39	2, 38	mpgbir 1831	. 2 ⊢ {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
40	1, 39	eqtri 2783	1 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3896   ⊊ wpss 4012   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148   class class class wbr 5102  Tr wtr 5211   I cid 5545   E cep 5550   × cxp 5649  ran crn 5652  Oncon0 6365   SSet csset 36690   Trans ctrans 36691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 8005  df-2nd 8006  df-txp 36712  df-sset 36714  df-trans 36715

This theorem is used by:  dfon4  36751