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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 31514 and bj-2upln0 31528 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 4844 |
. . . . . . 7
  
 tag 
tag  tag
tag | |
| 2 | 1 | difeq2i 3518 |
. . . . . 6
 tag ![\](setminus.gif "\"){kind=small}
tag
tag tag tag
tag |
| 3 | incom 3593 |
. . . . . . . . 9
tag
tag 
tag
tag
tag
tag | |
| 4 | bj-disjsn01 31454 |
. . . . . . . . . 10
| |
| 5 | xpdisj1 5215 |
. . . . . . . . . 10
 tag tag tag | |
| 6 | 4, 5 | ax-mp 5 |
. . . . . . . . 9
tag
tag
tag
|
| 7 | 3, 6 | eqtr3i 2447 |
. . . . . . . 8
tag
tag
tag |
| 8 | disjdif2 3814 |
. . . . . . . 8
tag tag
tag tag tag tag tag | |
| 9 | 7, 8 | ax-mp 5 |
. . . . . . 7
tag
tag
tag tag |
| 10 | bj-1ex 31455 |
. . . . . . . . . 10
  | |
| 11 | 10 | snnz 4056 |
. . . . . . . . 9
|
| 12 | bj-tagn0 31484 |
. . . . . . . . 9
tag | |
| 13 | 11, 12 | pm3.2i 456 |
. . . . . . . 8
![/\](wedge.gif "/\"){kind=small} tag |
| 14 | xpnz 5213 |
. . . . . . . 8
tag
tag
| |
| 15 | 13, 14 | mpbi 211 |
. . . . . . 7
tag |
| 16 | 9, 15 | eqnetri 2666 |
. . . . . 6
tag
tag
tag |
| 17 | 2, 16 | eqnetrri 2667 |
. . . . 5
tag tag
tag |
| 18 | 0pss 3770 |
. . . . 5
 tag tag tag
tag
tag
tag | |
| 19 | 17, 18 | mpbir 212 |
. . . 4
tag tag
tag |
| 20 | ssun2 3568 |
. . . . . . . 8
tag 
tag tag | |
| 21 | sscon 3537 |
. . . . . . . 8
tag
tag
tag
tag
tag
tag tag tag | |
| 22 | 20, 21 | ax-mp 5 |
. . . . . . 7
tag tag
tag tag tag |
| 23 | ssun2 3568 |
. . . . . . . 8
tag
tag tag | |
| 24 | ssdif 3538 |
. . . . . . . 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 3411 |
. . . . . 6
tag tag
tag tag tag tag |
| 27 | df-bj-2upl 31516 |
. . . . . . . 8
(|, |) (||) tag | |
| 28 | df-bj-1upl 31503 |
. . . . . . . . 9
(||) tag | |
| 29 | 28 | uneq1i 3554 |
. . . . . . . 8
(||) tag
tag tag |
| 30 | 27, 29 | eqtri 2445 |
. . . . . . 7
(|, |)
tag tag |
| 31 | 30 | difeq1i 3517 |
. . . . . 6
(|, |)
tag tag tag tag |
| 32 | 26, 31 | sseqtr4i 3435 |
. . . . 5
tag tag
tag (|, |)
tag |
| 33 | df-bj-1upl 31503 |
. . . . . 6
(||) tag | |
| 34 | 33 | difeq2i 3518 |
. . . . 5
(|, |)
(||) (|, |) tag |
| 35 | 32, 34 | sseqtr4i 3435 |
. . . 4
tag tag
tag (|, |)
(||) |
| 36 | psssstr 3509 |
. . . 4
tag
tag
tag tag tag
tag (|, |)
(||)
(|, |)
(||) | |
| 37 | 19, 35, 36 | mp2an 676 |
. . 3
(|, |)
(||) |
| 38 | 0pss 3770 |
. . 3
(|, |) (||)
(|, |) (||)
| |
| 39 | 37, 38 | mpbi 211 |
. 2
(|, |)
(||) |
| 40 | difn0 3792 |
. 2
(|, |) (||)
(|, |) (||) | |
| 41 | 39, 40 | ax-mp 5 |
1
(|, |) (||) |
| Colors of variables: wff setvar class |
| Syntax hints: wa 370
wceq 1437
wne 2594
cdif 3371
cun 3372
cin 3373
wss 3374
wpss 3375
c0 3699
csn 3936
cxp 4789
c1o 7125
tag bj-ctag 31479 (|bj-c1upl 31502 (|bj-c2uple 31515 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1663 ax-4 1676 ax-5 1752 ax-6 1798 ax-7 1843 ax-8 1874 ax-9 1876 ax-10 1891 ax-11 1896 ax-12 1909 ax-13 2058 ax-ext 2403 ax-sep 4484 ax-nul 4493 ax-pr 4598 ax-un 6536 |
| This theorem depends on definitions: df-bi 188 df-or 371 df-an 372 df-3an 984 df-tru 1440 df-ex 1658 df-nf 1662 df-sb 1791 df-eu 2275 df-mo 2276 df-clab 2410 df-cleq 2416 df-clel 2419 df-nfc 2553 df-ne 2596 df-ral 2714 df-rex 2715 df-rab 2718 df-v 3019 df-dif 3377 df-un 3379 df-in 3381 df-ss 3388 df-pss 3390 df-nul 3700 df-if 3850 df-sn 3937 df-pr 3939 df-op 3943 df-uni 4158 df-br 4362 df-opab 4421 df-xp 4797 df-rel 4798 df-cnv 4799 df-suc 5386 df-1o 7132 df-bj-tag 31480 df-bj-1upl 31503 df-bj-2upl 31516 |
| This theorem is referenced by: (None) |
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