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Theorem elirr 4204
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 3468 . . . . . . . . 9 ((A A y(y xy (V ∖ {A}))) → ¬ A (V ∖ {A}))
2 simp1 890 . . . . . . . . . . 11 ((A A y(y xy (V ∖ {A})) x = A) → A A)
3 eleq1 2078 . . . . . . . . . . . . . . . 16 (y = A → (y xA x))
4 eleq1 2078 . . . . . . . . . . . . . . . 16 (y = A → (y (V ∖ {A}) ↔ A (V ∖ {A})))
53, 4imbi12d 223 . . . . . . . . . . . . . . 15 (y = A → ((y xy (V ∖ {A})) ↔ (A xA (V ∖ {A}))))
65spcgv 2613 . . . . . . . . . . . . . 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 912 . . . . . . . . . . . 12 ((A A y(y xy (V ∖ {A})) x = A) → (A xA (V ∖ {A})))
9 eleq2 2079 . . . . . . . . . . . . . 14 (x = A → (A xA A))
109imbi1d 220 . . . . . . . . . . . . 13 (x = A → ((A xA (V ∖ {A})) ↔ (A AA (V ∖ {A}))))
11103ad2ant3 913 . . . . . . . . . . . 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 1090 . . . . . . . . 9 ((A A y(y xy (V ∖ {A}))) → (x = AA (V ∖ {A})))
151, 14mtod 576 . . . . . . . 8 ((A A y(y xy (V ∖ {A}))) → ¬ x = A)
16 vex 2534 . . . . . . . . . 10 x V
17 eldif 2900 . . . . . . . . . 10 (x (V ∖ {A}) ↔ (x V ¬ x {A}))
1816, 17mpbiran 833 . . . . . . . . 9 (x (V ∖ {A}) ↔ ¬ x {A})
19 elsn 3361 . . . . . . . . 9 (x {A} ↔ x = A)
2018, 19xchbinx 594 . . . . . . . 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 1732 . . . . 5 (A Ax(y(y xy (V ∖ {A})) → x (V ∖ {A})))
24 df-ral 2285 . . . . . . . 8 (y x [y / x]x (V ∖ {A}) ↔ y(y x → [y / x]x (V ∖ {A})))
25 clelsb3 2120 . . . . . . . . . 10 ([y / x]x (V ∖ {A}) ↔ y (V ∖ {A}))
2625imbi2i 215 . . . . . . . . 9 ((y x → [y / x]x (V ∖ {A})) ↔ (y xy (V ∖ {A})))
2726albii 1335 . . . . . . . 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 1335 . . . . 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 4200 . . . 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 2078 . . . 4 (x = A → (x (V ∖ {A}) ↔ A (V ∖ {A})))
3534spcgv 2613 . . 3 (A A → (x x (V ∖ {A}) → A (V ∖ {A})))
3633, 35mpd 13 . 2 (A AA (V ∖ {A}))
37 neldifsnd 3468 . 2 (A A → ¬ A (V ∖ {A}))
3836, 37pm2.65i 555 1 ¬ A A
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3a 871  wal 1224   = wceq 1226   wcel 1370  [wsb 1623  wral 2280  Vcvv 2531  cdif 2887  {csn 3346
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-v 2533  df-dif 2893  df-sn 3352
This theorem is referenced by:  ordirr  4205  elirrv  4206  sucprcreg  4207  dtruex  4217  ordsoexmid  4220  onnmin  4224  ssnel  4225  onpsssuc  4227  nntri2  5984  nndceq  5986  nndcel  5987
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