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Mirrors > Home > MPE Home > Th. List > epn0 | Structured version Visualization version GIF version |
Description: The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.) |
Ref | Expression |
---|---|
epn0 | ⊢ E ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sn0ep 5463 | . 2 ⊢ ∅ E {∅} | |
2 | brne0 5107 | . 2 ⊢ (∅ E {∅} → E ≠ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ E ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 3013 ∅c0 4288 {csn 4557 class class class wbr 5057 E cep 5457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-eprel 5458 |
This theorem is referenced by: epnsym 9060 |
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