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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2upln1upl | Structured 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 33666 and bj-2upln0 33680 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 5052 |
. . . . . . 7
  
 tag 
tag  tag
tag | |
| 2 | 1 | difeq2i 3619 |
. . . . . 6
 tag ![\](setminus.gif "\"){kind=small}
tag
tag tag tag
tag |
| 3 | incom 3691 |
. . . . . . . . 9
tag
tag 
tag
tag
tag
tag | |
| 4 | bj-disjsn01 33605 |
. . . . . . . . . 10
| |
| 5 | xpdisj1 5428 |
. . . . . . . . . 10
 tag tag tag | |
| 6 | 4, 5 | ax-mp 5 |
. . . . . . . . 9
tag
tag
tag
|
| 7 | 3, 6 | eqtr3i 2498 |
. . . . . . . 8
tag
tag
tag |
| 8 | bj-disjdif 33610 |
. . . . . . . 8
tag tag
tag tag tag tag tag | |
| 9 | 7, 8 | ax-mp 5 |
. . . . . . 7
tag
tag
tag tag |
| 10 | bj-1ex 33606 |
. . . . . . . . . 10
  | |
| 11 | 10 | snnz 4145 |
. . . . . . . . 9
|
| 12 | bj-tagn0 33636 |
. . . . . . . . 9
tag | |
| 13 | 11, 12 | pm3.2i 455 |
. . . . . . . 8
![/\](wedge.gif "/\"){kind=small} tag |
| 14 | xpnz 5426 |
. . . . . . . 8
tag
tag
| |
| 15 | 13, 14 | mpbi 208 |
. . . . . . 7
tag |
| 16 | 9, 15 | eqnetri 2763 |
. . . . . 6
tag
tag
tag |
| 17 | 2, 16 | eqnetrri 2764 |
. . . . 5
tag tag
tag |
| 18 | 0pss 3864 |
. . . . 5
 tag tag tag
tag
tag
tag | |
| 19 | 17, 18 | mpbir 209 |
. . . 4
tag tag
tag |
| 20 | ssun2 3668 |
. . . . . . . 8
tag 
tag tag | |
| 21 | sscon 3638 |
. . . . . . . 8
tag
tag
tag
tag
tag
tag tag tag | |
| 22 | 20, 21 | ax-mp 5 |
. . . . . . 7
tag tag
tag tag tag |
| 23 | ssun2 3668 |
. . . . . . . 8
tag
tag tag | |
| 24 | ssdif 3639 |
. . . . . . . 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 3513 |
. . . . . 6
tag tag
tag tag tag tag |
| 27 | df-bj-2upl 33668 |
. . . . . . . 8
(|, |) (||) tag | |
| 28 | df-bj-1upl 33655 |
. . . . . . . . 9
(||) tag | |
| 29 | 28 | uneq1i 3654 |
. . . . . . . 8
(||) tag
tag tag |
| 30 | 27, 29 | eqtri 2496 |
. . . . . . 7
(|, |)
tag tag |
| 31 | 30 | difeq1i 3618 |
. . . . . 6
(|, |)
tag tag tag tag |
| 32 | 26, 31 | sseqtr4i 3537 |
. . . . 5
tag tag
tag (|, |)
tag |
| 33 | df-bj-1upl 33655 |
. . . . . 6
(||) tag | |
| 34 | 33 | difeq2i 3619 |
. . . . 5
(|, |)
(||) (|, |) tag |
| 35 | 32, 34 | sseqtr4i 3537 |
. . . 4
tag tag
tag (|, |)
(||) |
| 36 | psssstr 3610 |
. . . 4
tag
tag
tag tag tag
tag (|, |)
(||)
(|, |)
(||) | |
| 37 | 19, 35, 36 | mp2an 672 |
. . 3
(|, |)
(||) |
| 38 | 0pss 3864 |
. . 3
(|, |) (||)
(|, |) (||)
| |
| 39 | 37, 38 | mpbi 208 |
. 2
(|, |)
(||) |
| 40 | difneqnul 27117 |
. 2
(|, |) (||)
(|, |) (||) | |
| 41 | 39, 40 | ax-mp 5 |
1
(|, |) (||) |
| Colors of variables: wff setvar class |
| Syntax hints: wa 369
wceq 1379
wne 2662
cdif 3473
cun 3474
cin 3475
wss 3476
wpss 3477
c0 3785
csn 4027
cxp 4997
c1o 7123
tag bj-ctag 33631 (|bj-c1upl 33654 (|bj-c2uple 33667 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-8 1769 ax-9 1771 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445 ax-sep 4568 ax-nul 4576 ax-pr 4686 ax-un 6576 |
| This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1382 df-ex 1597 df-nf 1600 df-sb 1712 df-eu 2279 df-mo 2280 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-ral 2819 df-rex 2820 df-rab 2823 df-v 3115 df-dif 3479 df-un 3481 df-in 3483 df-ss 3490 df-pss 3492 df-nul 3786 df-if 3940 df-sn 4028 df-pr 4030 df-op 4034 df-uni 4246 df-br 4448 df-opab 4506 df-suc 4884 df-xp 5005 df-rel 5006 df-cnv 5007 df-1o 7130 df-bj-tag 33632 df-bj-1upl 33655 df-bj-2upl 33668 |
| This theorem is referenced by: (None) |
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