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Theorem elirr 4224
Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.)
Assertion
Ref Expression
elirr ¬ A A

Proof of Theorem elirr
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neldifsnd 3489 . . . . . . . . 9 ((A A y(y xy (V ∖ {A}))) → ¬ A (V ∖ {A}))
2 simp1 903 . . . . . . . . . . 11 ((A A y(y xy (V ∖ {A})) x = A) → A A)
3 eleq1 2097 . . . . . . . . . . . . . . . 16 (y = A → (y xA x))
4 eleq1 2097 . . . . . . . . . . . . . . . 16 (y = A → (y (V ∖ {A}) ↔ A (V ∖ {A})))
53, 4imbi12d 223 . . . . . . . . . . . . . . 15 (y = A → ((y xy (V ∖ {A})) ↔ (A xA (V ∖ {A}))))
65spcgv 2634 . . . . . . . . . . . . . 14 (A x → (y(y xy (V ∖ {A})) → (A xA (V ∖ {A}))))
76pm2.43b 46 . . . . . . . . . . . . 13 (y(y xy (V ∖ {A})) → (A xA (V ∖ {A})))
873ad2ant2 925 . . . . . . . . . . . 12 ((A A y(y xy (V ∖ {A})) x = A) → (A xA (V ∖ {A})))
9 eleq2 2098 . . . . . . . . . . . . . 14 (x = A → (A xA A))
109imbi1d 220 . . . . . . . . . . . . 13 (x = A → ((A xA (V ∖ {A})) ↔ (A AA (V ∖ {A}))))
11103ad2ant3 926 . . . . . . . . . . . 12 ((A A y(y xy (V ∖ {A})) x = A) → ((A xA (V ∖ {A})) ↔ (A AA (V ∖ {A}))))
128, 11mpbid 135 . . . . . . . . . . 11 ((A A y(y xy (V ∖ {A})) x = A) → (A AA (V ∖ {A})))
132, 12mpd 13 . . . . . . . . . 10 ((A A y(y xy (V ∖ {A})) x = A) → A (V ∖ {A}))
14133expia 1105 . . . . . . . . 9 ((A A y(y xy (V ∖ {A}))) → (x = AA (V ∖ {A})))
151, 14mtod 588 . . . . . . . 8 ((A A y(y xy (V ∖ {A}))) → ¬ x = A)
16 vex 2554 . . . . . . . . . 10 x V
17 eldif 2921 . . . . . . . . . 10 (x (V ∖ {A}) ↔ (x V ¬ x {A}))
1816, 17mpbiran 846 . . . . . . . . 9 (x (V ∖ {A}) ↔ ¬ x {A})
19 elsn 3382 . . . . . . . . 9 (x {A} ↔ x = A)
2018, 19xchbinx 606 . . . . . . . 8 (x (V ∖ {A}) ↔ ¬ x = A)
2115, 20sylibr 137 . . . . . . 7 ((A A y(y xy (V ∖ {A}))) → x (V ∖ {A}))
2221ex 108 . . . . . 6 (A A → (y(y xy (V ∖ {A})) → x (V ∖ {A})))
2322alrimiv 1751 . . . . 5 (A Ax(y(y xy (V ∖ {A})) → x (V ∖ {A})))
24 df-ral 2305 . . . . . . . 8 (y x [y / x]x (V ∖ {A}) ↔ y(y x → [y / x]x (V ∖ {A})))
25 clelsb3 2139 . . . . . . . . . 10 ([y / x]x (V ∖ {A}) ↔ y (V ∖ {A}))
2625imbi2i 215 . . . . . . . . 9 ((y x → [y / x]x (V ∖ {A})) ↔ (y xy (V ∖ {A})))
2726albii 1356 . . . . . . . 8 (y(y x → [y / x]x (V ∖ {A})) ↔ y(y xy (V ∖ {A})))
2824, 27bitri 173 . . . . . . 7 (y x [y / x]x (V ∖ {A}) ↔ y(y xy (V ∖ {A})))
2928imbi1i 227 . . . . . 6 ((y x [y / x]x (V ∖ {A}) → x (V ∖ {A})) ↔ (y(y xy (V ∖ {A})) → x (V ∖ {A})))
3029albii 1356 . . . . 5 (x(y x [y / x]x (V ∖ {A}) → x (V ∖ {A})) ↔ x(y(y xy (V ∖ {A})) → x (V ∖ {A})))
3123, 30sylibr 137 . . . 4 (A Ax(y x [y / x]x (V ∖ {A}) → x (V ∖ {A})))
32 ax-setind 4220 . . . 4 (x(y x [y / x]x (V ∖ {A}) → x (V ∖ {A})) → x x (V ∖ {A}))
3331, 32syl 14 . . 3 (A Ax x (V ∖ {A}))
34 eleq1 2097 . . . 4 (x = A → (x (V ∖ {A}) ↔ A (V ∖ {A})))
3534spcgv 2634 . . 3 (A A → (x x (V ∖ {A}) → A (V ∖ {A})))
3633, 35mpd 13 . 2 (A AA (V ∖ {A}))
37 neldifsnd 3489 . 2 (A A → ¬ A (V ∖ {A}))
3836, 37pm2.65i 567 1 ¬ A A
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242   wcel 1390  [wsb 1642  wral 2300  Vcvv 2551  cdif 2908  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-sn 3373
This theorem is referenced by:  ordirr  4225  elirrv  4226  sucprcreg  4227  dtruex  4237  ordsoexmid  4240  onnmin  4244  ssnel  4245  onpsssuc  4247  nntri2  6012  nntri3  6014  nndceq  6015  nndcel  6016
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