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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. Theorem 1.9(i) of [Schloeder] p. 1. (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 2835 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) |
| 3 | 2 | notbid 318 | . . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴)) |
| 4 | elirrv 9636 | . . 3 ⊢ ¬ 𝑥 ∈ 𝑥 | |
| 5 | 3, 4 | vtoclg 3554 | . 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 1540 ∈ wcel 2108 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: elneq 9638 sucprcreg 9641 ruv 9642 disjcsn 9644 alephval3 10150 prv1n 35436 rankeq1o 36172 hfninf 36187 bj-iomnnom 37260 omabs2 43345 |
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