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Theorem dfon3 32329
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 32026 . 2 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
2 abeq1 2882 . . 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 3354 . . . . . . 7 𝑥 ∈ V
43elrn 5502 . . . . . 6 (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5 brin 4838 . . . . . . . . . . 11 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥𝑦( Trans × V)𝑥))
63brsset 32326 . . . . . . . . . . . 12 (𝑦 SSet 𝑥𝑦𝑥)
7 brxp 5285 . . . . . . . . . . . . . 14 (𝑦( Trans × V)𝑥 ↔ (𝑦 Trans 𝑥 ∈ V))
83, 7mpbiran2 689 . . . . . . . . . . . . 13 (𝑦( Trans × V)𝑥𝑦 Trans )
9 vex 3354 . . . . . . . . . . . . . 14 𝑦 ∈ V
109eltrans 32328 . . . . . . . . . . . . 13 (𝑦 Trans ↔ Tr 𝑦)
118, 10bitri 264 . . . . . . . . . . . 12 (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
126, 11anbi12i 612 . . . . . . . . . . 11 ((𝑦 SSet 𝑥𝑦( Trans × V)𝑥) ↔ (𝑦𝑥 ∧ Tr 𝑦))
135, 12bitri 264 . . . . . . . . . 10 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦𝑥 ∧ Tr 𝑦))
14 ioran 968 . . . . . . . . . . 11 (¬ (𝑦 = 𝑥𝑦𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
15 brun 4837 . . . . . . . . . . . 12 (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥𝑦 E 𝑥))
163ideq 5411 . . . . . . . . . . . . 13 (𝑦 I 𝑥𝑦 = 𝑥)
17 epel 5165 . . . . . . . . . . . . 13 (𝑦 E 𝑥𝑦𝑥)
1816, 17orbi12i 900 . . . . . . . . . . . 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 612 . . . . . . . . 9 ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
22 brdif 4839 . . . . . . . . 9 (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23 dfpss2 3842 . . . . . . . . . . . . 13 (𝑦𝑥 ↔ (𝑦𝑥 ∧ ¬ 𝑦 = 𝑥))
2423anbi1i 610 . . . . . . . . . . . 12 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25 an32 625 . . . . . . . . . . . 12 (((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2624, 25bitri 264 . . . . . . . . . . 11 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2726anbi1i 610 . . . . . . . . . 10 (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦𝑥))
28 anass 454 . . . . . . . . . 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 1924 . . . . . . 7 (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥))
32 exanali 1937 . . . . . . 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 3733 . . . . 5 (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
373, 36mpbiran 688 . . . 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 1874 . 2 {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
401, 39eqtri 2793 1 On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  wpss 3724   class class class wbr 4786  Tr wtr 4886   I cid 5156   E cep 5161   × cxp 5247  ran crn 5250  Oncon0 5864   SSet csset 32269   Trans ctrans 32270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ord 5867  df-on 5868  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fo 6035  df-fv 6037  df-1st 7313  df-2nd 7314  df-txp 32291  df-sset 32293  df-trans 32294
This theorem is referenced by:  dfon4  32330
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