Nearby theorems
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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 34968 and bj-2upln0 34982 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 5041 |
. . . . . . 7
  
 tag 
tag  tag
tag | |
| 2 | 1 | difeq2i 3605 |
. . . . . 6
 tag ![\](setminus.gif "\"){kind=small}
tag
tag tag tag
tag |
| 3 | incom 3677 |
. . . . . . . . 9
tag
tag 
tag
tag
tag
tag | |
| 4 | bj-disjsn01 34907 |
. . . . . . . . . 10
| |
| 5 | xpdisj1 5413 |
. . . . . . . . . 10
 tag tag tag | |
| 6 | 4, 5 | ax-mp 5 |
. . . . . . . . 9
tag
tag
tag
|
| 7 | 3, 6 | eqtr3i 2485 |
. . . . . . . 8
tag
tag
tag |
| 8 | bj-disjdif 34912 |
. . . . . . . 8
tag tag
tag tag tag tag tag | |
| 9 | 7, 8 | ax-mp 5 |
. . . . . . 7
tag
tag
tag tag |
| 10 | bj-1ex 34908 |
. . . . . . . . . 10
  | |
| 11 | 10 | snnz 4134 |
. . . . . . . . 9
|
| 12 | bj-tagn0 34938 |
. . . . . . . . 9
tag | |
| 13 | 11, 12 | pm3.2i 453 |
. . . . . . . 8
![/\](wedge.gif "/\"){kind=small} tag |
| 14 | xpnz 5411 |
. . . . . . . 8
tag
tag
| |
| 15 | 13, 14 | mpbi 208 |
. . . . . . 7
tag |
| 16 | 9, 15 | eqnetri 2750 |
. . . . . 6
tag
tag
tag |
| 17 | 2, 16 | eqnetrri 2751 |
. . . . 5
tag tag
tag |
| 18 | 0pss 3852 |
. . . . 5
 tag tag tag
tag
tag
tag | |
| 19 | 17, 18 | mpbir 209 |
. . . 4
tag tag
tag |
| 20 | ssun2 3654 |
. . . . . . . 8
tag 
tag tag | |
| 21 | sscon 3624 |
. . . . . . . 8
tag
tag
tag
tag
tag
tag tag tag | |
| 22 | 20, 21 | ax-mp 5 |
. . . . . . 7
tag tag
tag tag tag |
| 23 | ssun2 3654 |
. . . . . . . 8
tag
tag tag | |
| 24 | ssdif 3625 |
. . . . . . . 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 3498 |
. . . . . 6
tag tag
tag tag tag tag |
| 27 | df-bj-2upl 34970 |
. . . . . . . 8
(|, |) (||) tag | |
| 28 | df-bj-1upl 34957 |
. . . . . . . . 9
(||) tag | |
| 29 | 28 | uneq1i 3640 |
. . . . . . . 8
(||) tag
tag tag |
| 30 | 27, 29 | eqtri 2483 |
. . . . . . 7
(|, |)
tag tag |
| 31 | 30 | difeq1i 3604 |
. . . . . 6
(|, |)
tag tag tag tag |
| 32 | 26, 31 | sseqtr4i 3522 |
. . . . 5
tag tag
tag (|, |)
tag |
| 33 | df-bj-1upl 34957 |
. . . . . 6
(||) tag | |
| 34 | 33 | difeq2i 3605 |
. . . . 5
(|, |)
(||) (|, |) tag |
| 35 | 32, 34 | sseqtr4i 3522 |
. . . 4
tag tag
tag (|, |)
(||) |
| 36 | psssstr 3596 |
. . . 4
tag
tag
tag tag tag
tag (|, |)
(||)
(|, |)
(||) | |
| 37 | 19, 35, 36 | mp2an 670 |
. . 3
(|, |)
(||) |
| 38 | 0pss 3852 |
. . 3
(|, |) (||)
(|, |) (||)
| |
| 39 | 37, 38 | mpbi 208 |
. 2
(|, |)
(||) |
| 40 | difneqnul 27614 |
. 2
(|, |) (||)
(|, |) (||) | |
| 41 | 39, 40 | ax-mp 5 |
1
(|, |) (||) |
| Colors of variables: wff setvar class |
| Syntax hints: wa 367
wceq 1398
wne 2649
cdif 3458
cun 3459
cin 3460
wss 3461
wpss 3462
c0 3783
csn 4016
cxp 4986
c1o 7115
tag bj-ctag 34933 (|bj-c1upl 34956 (|bj-c2uple 34969 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1623 ax-4 1636 ax-5 1709 ax-6 1752 ax-7 1795 ax-8 1825 ax-9 1827 ax-10 1842 ax-11 1847 ax-12 1859 ax-13 2004 ax-ext 2432 ax-sep 4560 ax-nul 4568 ax-pr 4676 ax-un 6565 |
| This theorem depends on definitions: df-bi 185 df-or 368 df-an 369 df-3an 973 df-tru 1401 df-ex 1618 df-nf 1622 df-sb 1745 df-eu 2288 df-mo 2289 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2651 df-ral 2809 df-rex 2810 df-rab 2813 df-v 3108 df-dif 3464 df-un 3466 df-in 3468 df-ss 3475 df-pss 3477 df-nul 3784 df-if 3930 df-sn 4017 df-pr 4019 df-op 4023 df-uni 4236 df-br 4440 df-opab 4498 df-suc 4873 df-xp 4994 df-rel 4995 df-cnv 4996 df-1o 7122 df-bj-tag 34934 df-bj-1upl 34957 df-bj-2upl 34970 |
| This theorem is referenced by: (None) |
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