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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 2831 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) |
3 | 2 | notbid 319 | . . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴)) |
4 | elirrv 9395 | . . 3 ⊢ ¬ 𝑥 ∈ 𝑥 | |
5 | 3, 4 | vtoclg 3510 | . 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴) |
6 | pm2.01 188 | . 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 1539 ∈ wcel 2104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-reg 9391 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-v 3439 df-dif 3895 df-un 3897 df-nul 4263 df-sn 4566 df-pr 4568 |
This theorem is referenced by: elneq 9397 sucprcreg 9400 ruv 9401 alephval3 9908 bnj521 32757 prv1n 33434 rankeq1o 34514 hfninf 34529 bj-disjcsn 35182 bj-iomnnom 35471 |
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