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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 9062 | . 2 ⊢ ¬ 𝑦 ∈ 𝑦 | |
2 | 1 | nfth 1802 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦 |
3 | ax8 2120 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦)) | |
4 | 3 | con3d 155 | . . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦)) |
5 | 2, 4 | spime 2407 | . 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 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-reg 9058 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 |
This theorem is referenced by: (None) |
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