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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1upln0 | Structured version Visualization version GIF version |
Description: A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-1upln0 | ⊢ ⦅𝐴⦆ ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-1upl 35868 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
2 | 0nep0 5356 | . . . 4 ⊢ ∅ ≠ {∅} | |
3 | 2 | necomi 2996 | . . 3 ⊢ {∅} ≠ ∅ |
4 | bj-tagn0 35849 | . . 3 ⊢ tag 𝐴 ≠ ∅ | |
5 | xpnz 6156 | . . . 4 ⊢ (({∅} ≠ ∅ ∧ tag 𝐴 ≠ ∅) ↔ ({∅} × tag 𝐴) ≠ ∅) | |
6 | 5 | biimpi 215 | . . 3 ⊢ (({∅} ≠ ∅ ∧ tag 𝐴 ≠ ∅) → ({∅} × tag 𝐴) ≠ ∅) |
7 | 3, 4, 6 | mp2an 691 | . 2 ⊢ ({∅} × tag 𝐴) ≠ ∅ |
8 | 1, 7 | eqnetri 3012 | 1 ⊢ ⦅𝐴⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ≠ wne 2941 ∅c0 4322 {csn 4628 × cxp 5674 tag bj-ctag 35844 ⦅bj-c1upl 35867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-bj-tag 35845 df-bj-1upl 35868 |
This theorem is referenced by: bj-2upln0 35893 |
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