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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 37000 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
| 2 | 0nep0 5357 | . . . 4 ⊢ ∅ ≠ {∅} | |
| 3 | 2 | necomi 2994 | . . 3 ⊢ {∅} ≠ ∅ | 
| 4 | bj-tagn0 36981 | . . 3 ⊢ tag 𝐴 ≠ ∅ | |
| 5 | xpnz 6178 | . . . 4 ⊢ (({∅} ≠ ∅ ∧ tag 𝐴 ≠ ∅) ↔ ({∅} × tag 𝐴) ≠ ∅) | |
| 6 | 5 | biimpi 216 | . . 3 ⊢ (({∅} ≠ ∅ ∧ tag 𝐴 ≠ ∅) → ({∅} × tag 𝐴) ≠ ∅) | 
| 7 | 3, 4, 6 | mp2an 692 | . 2 ⊢ ({∅} × tag 𝐴) ≠ ∅ | 
| 8 | 1, 7 | eqnetri 3010 | 1 ⊢ ⦅𝐴⦆ ≠ ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ≠ wne 2939 ∅c0 4332 {csn 4625 × cxp 5682 tag bj-ctag 36976 ⦅bj-c1upl 36999 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-bj-tag 36977 df-bj-1upl 37000 | 
| This theorem is referenced by: bj-2upln0 37025 | 
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