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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > exnel | Structured version Visualization version GIF version |
Description: There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.) |
Ref | Expression |
---|---|
exnel | ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 9619 | . 2 ⊢ ¬ 𝑦 ∈ 𝑦 | |
2 | 1 | nfth 1796 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦 |
3 | ax8 2105 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦)) | |
4 | 3 | con3d 152 | . . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦)) |
5 | 2, 4 | spime 2384 | . 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-13 2367 ax-ext 2699 ax-sep 5299 ax-pr 5429 ax-reg 9615 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-v 3473 df-un 3952 df-sn 4630 df-pr 4632 |
This theorem is referenced by: (None) |
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