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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2upln1upl | Structured version Visualization version Unicode version | ||
Description: A couple is never equal
to a monuple. It is in order to have this
"non-clashing" result that tagging was used. Without tagging,
we would
have (|   |)  (||). Note that in the context of Morse
tuples, it is natural to define the 0-tuple as the empty set. Therefore,
the present theorem together with bj-1upln0 31647 and bj-2upln0 31661 tell us that
an m-tuple may equal an n-tuple only when m = n, at least for m, n <=
2,
but this result would extend as soon as we define n-tuples for higher
values of n. (Contributed by BJ, 21-Apr-2019.) |
| Ref | Expression |
|---|---|
| bj-2upln1upl |
 (|,  |)  (| |) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xpundi 4905 |
. . . . . . 7
  
 tag 
tag  tag
tag | |
| 2 | 1 | difeq2i 3559 |
. . . . . 6
 tag ![\](setminus.gif "\"){kind=small}
tag
tag tag tag
tag |
| 3 | incom 3636 |
. . . . . . . . 9
tag
tag 
tag
tag
tag
tag | |
| 4 | bj-disjsn01 31587 |
. . . . . . . . . 10
| |
| 5 | xpdisj1 5276 |
. . . . . . . . . 10
 tag tag tag | |
| 6 | 4, 5 | ax-mp 5 |
. . . . . . . . 9
tag
tag
tag
|
| 7 | 3, 6 | eqtr3i 2485 |
. . . . . . . 8
tag
tag
tag |
| 8 | disjdif2 3857 |
. . . . . . . 8
tag tag
tag tag tag tag tag | |
| 9 | 7, 8 | ax-mp 5 |
. . . . . . 7
tag
tag
tag tag |
| 10 | bj-1ex 31588 |
. . . . . . . . . 10
  | |
| 11 | 10 | snnz 4102 |
. . . . . . . . 9
|
| 12 | bj-tagn0 31617 |
. . . . . . . . 9
tag | |
| 13 | 11, 12 | pm3.2i 461 |
. . . . . . . 8
![/\](wedge.gif "/\"){kind=small} tag |
| 14 | xpnz 5274 |
. . . . . . . 8
tag
tag
| |
| 15 | 13, 14 | mpbi 213 |
. . . . . . 7
tag |
| 16 | 9, 15 | eqnetri 2705 |
. . . . . 6
tag
tag
tag |
| 17 | 2, 16 | eqnetrri 2706 |
. . . . 5
tag tag
tag |
| 18 | 0pss 3813 |
. . . . 5
 tag tag tag
tag
tag
tag | |
| 19 | 17, 18 | mpbir 214 |
. . . 4
tag tag
tag |
| 20 | ssun2 3609 |
. . . . . . . 8
tag 
tag tag | |
| 21 | sscon 3578 |
. . . . . . . 8
tag
tag
tag
tag
tag
tag tag tag | |
| 22 | 20, 21 | ax-mp 5 |
. . . . . . 7
tag tag
tag tag tag |
| 23 | ssun2 3609 |
. . . . . . . 8
tag
tag tag | |
| 24 | ssdif 3579 |
. . . . . . . 8
tag
tag tag tag tag tag tag tag | |
| 25 | 23, 24 | ax-mp 5 |
. . . . . . 7
tag
tag
tag tag tag |
| 26 | 22, 25 | sstri 3452 |
. . . . . 6
tag tag
tag tag tag tag |
| 27 | df-bj-2upl 31649 |
. . . . . . . 8
(|, |) (||) tag | |
| 28 | df-bj-1upl 31636 |
. . . . . . . . 9
(||) tag | |
| 29 | 28 | uneq1i 3595 |
. . . . . . . 8
(||) tag
tag tag |
| 30 | 27, 29 | eqtri 2483 |
. . . . . . 7
(|, |)
tag tag |
| 31 | 30 | difeq1i 3558 |
. . . . . 6
(|, |)
tag tag tag tag |
| 32 | 26, 31 | sseqtr4i 3476 |
. . . . 5
tag tag
tag (|, |)
tag |
| 33 | df-bj-1upl 31636 |
. . . . . 6
(||) tag | |
| 34 | 33 | difeq2i 3559 |
. . . . 5
(|, |)
(||) (|, |) tag |
| 35 | 32, 34 | sseqtr4i 3476 |
. . . 4
tag tag
tag (|, |)
(||) |
| 36 | psssstr 3550 |
. . . 4
tag
tag
tag tag tag
tag (|, |)
(||)
(|, |)
(||) | |
| 37 | 19, 35, 36 | mp2an 683 |
. . 3
(|, |)
(||) |
| 38 | 0pss 3813 |
. . 3
(|, |) (||)
(|, |) (||)
| |
| 39 | 37, 38 | mpbi 213 |
. 2
(|, |)
(||) |
| 40 | difn0 3835 |
. 2
(|, |) (||)
(|, |) (||) | |
| 41 | 39, 40 | ax-mp 5 |
1
(|, |) (||) |
| Colors of variables: wff setvar class |
| Syntax hints: wa 375
wceq 1454
wne 2632
cdif 3412
cun 3413
cin 3414
wss 3415
wpss 3416
c0 3742
csn 3979
cxp 4850
c1o 7200
tag bj-ctag 31612 (|bj-c1upl 31635 (|bj-c2uple 31648 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pr 4652 ax-un 6609 |
| This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-sn 3980 df-pr 3982 df-op 3986 df-uni 4212 df-br 4416 df-opab 4475 df-xp 4858 df-rel 4859 df-cnv 4860 df-suc 5447 df-1o 7207 df-bj-tag 31613 df-bj-1upl 31636 df-bj-2upl 31649 |
| This theorem is referenced by: (None) |
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