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Mirrors > Home > MPE Home > Th. List > elirr | Structured version Visualization version GIF version |
Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
elirr | ⊢ ¬ 𝐴 ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
2 | 1, 1 | eleq12d 2853 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) |
3 | 2 | notbid 310 | . . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴)) |
4 | elirrv 8790 | . . 3 ⊢ ¬ 𝑥 ∈ 𝑥 | |
5 | 3, 4 | vtoclg 3467 | . 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴) |
6 | pm2.01 181 | . 2 ⊢ ((𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-reg 8786 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-v 3400 df-dif 3795 df-un 3797 df-nul 4142 df-sn 4399 df-pr 4401 |
This theorem is referenced by: elneq 8792 sucprcreg 8795 alephval3 9266 n0lplig 27910 bnj521 31405 rankeq1o 32867 hfninf 32882 bj-disjcsn 33509 bj-iomnnom 33737 |
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