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Theorem efrirr 4090
 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 4087 . . . 4 (( E Fr 𝐴𝐴𝐴𝐴𝐴) → ¬ 𝐴 E 𝐴)
213anidm23 1194 . . 3 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴 E 𝐴)
3 epelg 4027 . . . 4 (𝐴𝐴 → (𝐴 E 𝐴𝐴𝐴))
43adantl 262 . . 3 (( E Fr 𝐴𝐴𝐴) → (𝐴 E 𝐴𝐴𝐴))
52, 4mtbid 597 . 2 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴𝐴)
65pm2.01da 565 1 ( E Fr 𝐴 → ¬ 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393   class class class wbr 3764   E cep 4024   Fr wfr 4065 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-frfor 4068  df-frind 4069 This theorem is referenced by:  tz7.2  4091
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