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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 2884 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴)) |
3 | 2 | notbid 321 | . . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴)) |
4 | elirrv 9044 | . . 3 ⊢ ¬ 𝑥 ∈ 𝑥 | |
5 | 3, 4 | vtoclg 3515 | . 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴) |
6 | pm2.01 192 | . 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 1538 ∈ wcel 2111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-nul 4244 df-sn 4526 df-pr 4528 |
This theorem is referenced by: elneq 9046 sucprcreg 9049 alephval3 9521 bnj521 32117 prv1n 32791 rankeq1o 33745 hfninf 33760 bj-disjcsn 34387 bj-iomnnom 34674 |
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