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Theorem dfon3 31668
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.
StepHypRef Expression
1 dfon2 31425 . 2 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
2 abeq1 2730 . . 3 ({𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ∀𝑥(∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))))
3 vex 3192 . . . . . . 7 𝑥 ∈ V
43elrn 5331 . . . . . 6 (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5 brin 4669 . . . . . . . . . . 11 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥𝑦( Trans × V)𝑥))
63brsset 31665 . . . . . . . . . . . 12 (𝑦 SSet 𝑥𝑦𝑥)
7 brxp 5112 . . . . . . . . . . . . . 14 (𝑦( Trans × V)𝑥 ↔ (𝑦 Trans 𝑥 ∈ V))
83, 7mpbiran2 953 . . . . . . . . . . . . 13 (𝑦( Trans × V)𝑥𝑦 Trans )
9 vex 3192 . . . . . . . . . . . . . 14 𝑦 ∈ V
109eltrans 31667 . . . . . . . . . . . . 13 (𝑦 Trans ↔ Tr 𝑦)
118, 10bitri 264 . . . . . . . . . . . 12 (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
126, 11anbi12i 732 . . . . . . . . . . 11 ((𝑦 SSet 𝑥𝑦( Trans × V)𝑥) ↔ (𝑦𝑥 ∧ Tr 𝑦))
135, 12bitri 264 . . . . . . . . . 10 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦𝑥 ∧ Tr 𝑦))
14 ioran 511 . . . . . . . . . . 11 (¬ (𝑦 = 𝑥𝑦𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
15 brun 4668 . . . . . . . . . . . 12 (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥𝑦 E 𝑥))
163ideq 5239 . . . . . . . . . . . . 13 (𝑦 I 𝑥𝑦 = 𝑥)
17 epel 4993 . . . . . . . . . . . . 13 (𝑦 E 𝑥𝑦𝑥)
1816, 17orbi12i 543 . . . . . . . . . . . 12 ((𝑦 I 𝑥𝑦 E 𝑥) ↔ (𝑦 = 𝑥𝑦𝑥))
1915, 18bitri 264 . . . . . . . . . . 11 (𝑦( I ∪ E )𝑥 ↔ (𝑦 = 𝑥𝑦𝑥))
2014, 19xchnxbir 323 . . . . . . . . . 10 𝑦( I ∪ E )𝑥 ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
2113, 20anbi12i 732 . . . . . . . . 9 ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
22 brdif 4670 . . . . . . . . 9 (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23 dfpss2 3675 . . . . . . . . . . . . 13 (𝑦𝑥 ↔ (𝑦𝑥 ∧ ¬ 𝑦 = 𝑥))
2423anbi1i 730 . . . . . . . . . . . 12 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25 an32 838 . . . . . . . . . . . 12 (((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2624, 25bitri 264 . . . . . . . . . . 11 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2726anbi1i 730 . . . . . . . . . 10 (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦𝑥))
28 anass 680 . . . . . . . . . 10 ((((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
2927, 28bitri 264 . . . . . . . . 9 (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
3021, 22, 293bitr4i 292 . . . . . . . 8 (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥))
3130exbii 1771 . . . . . . 7 (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥))
32 exanali 1783 . . . . . . 7 (∃𝑦((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ ¬ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
3331, 32bitri 264 . . . . . 6 (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ¬ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
344, 33bitri 264 . . . . 5 (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ¬ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
3534con2bii 347 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
36 eldif 3569 . . . . 5 (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
373, 36mpbiran 952 . . . 4 (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
3835, 37bitr4i 267 . . 3 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
392, 38mpgbir 1723 . 2 {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
401, 39eqtri 2643 1 On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  wpss 3560   class class class wbr 4618  Tr wtr 4717   E cep 4988   I cid 4989   × cxp 5077  ran crn 5080  Oncon0 5687   SSet csset 31607   Trans ctrans 31608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31629  df-sset 31631  df-trans 31632
This theorem is referenced by:  dfon4  31669
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