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Theorem dfon3 32963
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 32646 . 2 On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}
2 abeq1 2915 . . 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 3440 . . . . . . 7 𝑥 ∈ V
43elrn 5704 . . . . . 6 (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥)
5 brin 5014 . . . . . . . . . . 11 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦 SSet 𝑥𝑦( Trans × V)𝑥))
63brsset 32960 . . . . . . . . . . . 12 (𝑦 SSet 𝑥𝑦𝑥)
7 brxp 5489 . . . . . . . . . . . . . 14 (𝑦( Trans × V)𝑥 ↔ (𝑦 Trans 𝑥 ∈ V))
83, 7mpbiran2 706 . . . . . . . . . . . . 13 (𝑦( Trans × V)𝑥𝑦 Trans )
9 vex 3440 . . . . . . . . . . . . . 14 𝑦 ∈ V
109eltrans 32962 . . . . . . . . . . . . 13 (𝑦 Trans ↔ Tr 𝑦)
118, 10bitri 276 . . . . . . . . . . . 12 (𝑦( Trans × V)𝑥 ↔ Tr 𝑦)
126, 11anbi12i 626 . . . . . . . . . . 11 ((𝑦 SSet 𝑥𝑦( Trans × V)𝑥) ↔ (𝑦𝑥 ∧ Tr 𝑦))
135, 12bitri 276 . . . . . . . . . 10 (𝑦( SSet ∩ ( Trans × V))𝑥 ↔ (𝑦𝑥 ∧ Tr 𝑦))
14 ioran 978 . . . . . . . . . . 11 (¬ (𝑦 = 𝑥𝑦𝑥) ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
15 brun 5013 . . . . . . . . . . . 12 (𝑦( I ∪ E )𝑥 ↔ (𝑦 I 𝑥𝑦 E 𝑥))
163ideq 5609 . . . . . . . . . . . . 13 (𝑦 I 𝑥𝑦 = 𝑥)
17 epel 5357 . . . . . . . . . . . . 13 (𝑦 E 𝑥𝑦𝑥)
1816, 17orbi12i 909 . . . . . . . . . . . 12 ((𝑦 I 𝑥𝑦 E 𝑥) ↔ (𝑦 = 𝑥𝑦𝑥))
1915, 18bitri 276 . . . . . . . . . . 11 (𝑦( I ∪ E )𝑥 ↔ (𝑦 = 𝑥𝑦𝑥))
2014, 19xchnxbir 334 . . . . . . . . . 10 𝑦( I ∪ E )𝑥 ↔ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥))
2113, 20anbi12i 626 . . . . . . . . 9 ((𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
22 brdif 5015 . . . . . . . . 9 (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ (𝑦( SSet ∩ ( Trans × V))𝑥 ∧ ¬ 𝑦( I ∪ E )𝑥))
23 dfpss2 3983 . . . . . . . . . . . . 13 (𝑦𝑥 ↔ (𝑦𝑥 ∧ ¬ 𝑦 = 𝑥))
2423anbi1i 623 . . . . . . . . . . . 12 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦))
25 an32 642 . . . . . . . . . . . 12 (((𝑦𝑥 ∧ ¬ 𝑦 = 𝑥) ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2624, 25bitri 276 . . . . . . . . . . 11 ((𝑦𝑥 ∧ Tr 𝑦) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥))
2726anbi1i 623 . . . . . . . . . 10 (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦𝑥))
28 anass 469 . . . . . . . . . 10 ((((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦 = 𝑥) ∧ ¬ 𝑦𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
2927, 28bitri 276 . . . . . . . . 9 (((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ (¬ 𝑦 = 𝑥 ∧ ¬ 𝑦𝑥)))
3021, 22, 293bitr4i 304 . . . . . . . 8 (𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥))
3130exbii 1829 . . . . . . 7 (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ∃𝑦((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥))
32 exanali 1840 . . . . . . 7 (∃𝑦((𝑦𝑥 ∧ Tr 𝑦) ∧ ¬ 𝑦𝑥) ↔ ¬ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
3331, 32bitri 276 . . . . . 6 (∃𝑦 𝑦(( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))𝑥 ↔ ¬ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
344, 33bitri 276 . . . . 5 (𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) ↔ ¬ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥))
3534con2bii 359 . . . 4 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
36 eldif 3869 . . . . 5 (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
373, 36mpbiran 705 . . . 4 (𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) ↔ ¬ 𝑥 ∈ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
3835, 37bitr4i 279 . . 3 (∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) ↔ 𝑥 ∈ (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))))
392, 38mpgbir 1781 . 2 {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)} = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
401, 39eqtri 2819 1 On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842  wal 1520   = wceq 1522  wex 1761  wcel 2081  {cab 2775  Vcvv 3437  cdif 3856  cun 3857  cin 3858  wss 3859  wpss 3860   class class class wbr 4962  Tr wtr 5063   I cid 5347   E cep 5352   × cxp 5441  ran crn 5444  Oncon0 6066   SSet csset 32903   Trans ctrans 32904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ord 6069  df-on 6070  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fo 6231  df-fv 6233  df-1st 7545  df-2nd 7546  df-txp 32925  df-sset 32927  df-trans 32928
This theorem is referenced by:  dfon4  32964
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