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Mirrors > Home > MPE Home > Th. List > 0inp0 | Structured version Visualization version GIF version |
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
0inp0 | ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nep0 4836 | . . 3 ⊢ ∅ ≠ {∅} | |
2 | neeq1 2856 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅})) | |
3 | 1, 2 | mpbiri 248 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅}) |
4 | 3 | neneqd 2799 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ≠ wne 2794 ∅c0 3915 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-nul 3916 df-sn 4178 |
This theorem is referenced by: dtruALT 4899 zfpair 4904 dtruALT2 4911 |
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