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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 34434 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
2 | 0nep0 5223 | . . . 4 ⊢ ∅ ≠ {∅} | |
3 | 2 | necomi 3041 | . . 3 ⊢ {∅} ≠ ∅ |
4 | bj-tagn0 34415 | . . 3 ⊢ tag 𝐴 ≠ ∅ | |
5 | xpnz 5983 | . . . 4 ⊢ (({∅} ≠ ∅ ∧ tag 𝐴 ≠ ∅) ↔ ({∅} × tag 𝐴) ≠ ∅) | |
6 | 5 | biimpi 219 | . . 3 ⊢ (({∅} ≠ ∅ ∧ tag 𝐴 ≠ ∅) → ({∅} × tag 𝐴) ≠ ∅) |
7 | 3, 4, 6 | mp2an 691 | . 2 ⊢ ({∅} × tag 𝐴) ≠ ∅ |
8 | 1, 7 | eqnetri 3057 | 1 ⊢ ⦅𝐴⦆ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ≠ wne 2987 ∅c0 4243 {csn 4525 × cxp 5517 tag bj-ctag 34410 ⦅bj-c1upl 34433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-bj-tag 34411 df-bj-1upl 34434 |
This theorem is referenced by: bj-2upln0 34459 |
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