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Theorem efrirr 4116
Description: Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
efrirr ( E Fr 𝐴 → ¬ 𝐴𝐴)

Proof of Theorem efrirr
StepHypRef Expression
1 frirrg 4113 . . . 4 (( E Fr 𝐴𝐴𝐴𝐴𝐴) → ¬ 𝐴 E 𝐴)
213anidm23 1229 . . 3 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴 E 𝐴)
3 epelg 4053 . . . 4 (𝐴𝐴 → (𝐴 E 𝐴𝐴𝐴))
43adantl 271 . . 3 (( E Fr 𝐴𝐴𝐴) → (𝐴 E 𝐴𝐴𝐴))
52, 4mtbid 630 . 2 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴𝐴)
65pm2.01da 598 1 ( E Fr 𝐴 → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wcel 1434   class class class wbr 3793   E cep 4050   Fr wfr 4091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-eprel 4052  df-frfor 4094  df-frind 4095
This theorem is referenced by:  tz7.2  4117
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