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Theorem dfon3 32303
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 32000 . 2 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
2 abeq1 2869 . . 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 3341 . . . . . . 7 𝑥 ∈ V
43elrn 5519 . . . . . 6 (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5 brin 4854 . . . . . . . . . . 11 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥𝑦( Trans × V)𝑥))
63brsset 32300 . . . . . . . . . . . 12 (𝑦 SSet 𝑥𝑦𝑥)
7 brxp 5302 . . . . . . . . . . . . . 14 (𝑦( Trans × V)𝑥 ↔ (𝑦 Trans 𝑥 ∈ V))
83, 7mpbiran2 992 . . . . . . . . . . . . 13 (𝑦( Trans × V)𝑥𝑦 Trans )
9 vex 3341 . . . . . . . . . . . . . 14 𝑦 ∈ V
109eltrans 32302 . . . . . . . . . . . . 13 (𝑦 Trans ↔ Tr 𝑦)
118, 10bitri 264 . . . . . . . . . . . 12 (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
126, 11anbi12i 735 . . . . . . . . . . 11 ((𝑦 SSet 𝑥𝑦( Trans × V)𝑥) ↔ (𝑦𝑥 ∧ Tr 𝑦))
135, 12bitri 264 . . . . . . . . . 10 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦𝑥 ∧ Tr 𝑦))
14 ioran 512 . . . . . . . . . . 11 (¬ (𝑦 = 𝑥𝑦𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
15 brun 4853 . . . . . . . . . . . 12 (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥𝑦 E 𝑥))
163ideq 5428 . . . . . . . . . . . . 13 (𝑦 I 𝑥𝑦 = 𝑥)
17 epel 5180 . . . . . . . . . . . . 13 (𝑦 E 𝑥𝑦𝑥)
1816, 17orbi12i 544 . . . . . . . . . . . 12 ((𝑦 I 𝑥𝑦 E 𝑥) ↔ (𝑦 = 𝑥𝑦𝑥))
1915, 18bitri 264 . . . . . . . . . . 11 (𝑦( I ∪ E )𝑥 ↔ (𝑦 = 𝑥𝑦𝑥))
2014, 19xchnxbir 322 . . . . . . . . . 10 𝑦( I ∪ E )𝑥 ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
2113, 20anbi12i 735 . . . . . . . . 9 ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
22 brdif 4855 . . . . . . . . 9 (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23 dfpss2 3832 . . . . . . . . . . . . 13 (𝑦𝑥 ↔ (𝑦𝑥 ∧ ¬ 𝑦 = 𝑥))
2423anbi1i 733 . . . . . . . . . . . 12 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25 an32 874 . . . . . . . . . . . 12 (((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2624, 25bitri 264 . . . . . . . . . . 11 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2726anbi1i 733 . . . . . . . . . 10 (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦𝑥))
28 anass 684 . . . . . . . . . 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 1921 . . . . . . 7 (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥))
32 exanali 1933 . . . . . . 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 346 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
36 eldif 3723 . . . . 5 (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
373, 36mpbiran 991 . . . 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 1873 . 2 {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
401, 39eqtri 2780 1 On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wal 1628   = wceq 1630  wex 1851  wcel 2137  {cab 2744  Vcvv 3338  cdif 3710  cun 3711  cin 3712  wss 3713  wpss 3714   class class class wbr 4802  Tr wtr 4902   I cid 5171   E cep 5176   × cxp 5262  ran crn 5265  Oncon0 5882   SSet csset 32243   Trans ctrans 32244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ord 5885  df-on 5886  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fo 6053  df-fv 6055  df-1st 7331  df-2nd 7332  df-txp 32265  df-sset 32267  df-trans 32268
This theorem is referenced by:  dfon4  32304
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