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Theorem bj-2upln1upl 31529
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 (| A ,  (/)|)  = (| A|). 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.)
Assertion
Ref Expression
bj-2upln1upl  |- (| A,  B|)  =/= (| C|)

Proof of Theorem bj-2upln1upl
StepHypRef Expression
1 xpundi 4844 . . . . . . 7  |-  ( {
(/) }  X.  (tag  A  u. tag  C )
)  =  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) )
21difeq2i 3518 . . . . . 6  |-  ( ( { 1o }  X. tag  B )  \  ( {
(/) }  X.  (tag  A  u. tag  C )
) )  =  ( ( { 1o }  X. tag  B )  \  (
( { (/) }  X. tag  A )  u.  ( {
(/) }  X. tag  C ) ) )
3 incom 3593 . . . . . . . . 9  |-  ( ( { (/) }  X.  (tag  A  u. tag  C )
)  i^i  ( { 1o }  X. tag  B )
)  =  ( ( { 1o }  X. tag  B )  i^i  ( {
(/) }  X.  (tag  A  u. tag  C )
) )
4 bj-disjsn01 31454 . . . . . . . . . 10  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
5 xpdisj1 5215 . . . . . . . . . 10  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  ->  ( ( { (/) }  X.  (tag  A  u. tag  C ) )  i^i  ( { 1o }  X. tag  B
) )  =  (/) )
64, 5ax-mp 5 . . . . . . . . 9  |-  ( ( { (/) }  X.  (tag  A  u. tag  C )
)  i^i  ( { 1o }  X. tag  B )
)  =  (/)
73, 6eqtr3i 2447 . . . . . . . 8  |-  ( ( { 1o }  X. tag  B )  i^i  ( {
(/) }  X.  (tag  A  u. tag  C )
) )  =  (/)
8 disjdif2 3814 . . . . . . . 8  |-  ( ( ( { 1o }  X. tag  B )  i^i  ( { (/) }  X.  (tag  A  u. tag  C )
) )  =  (/)  ->  ( ( { 1o }  X. tag  B )  \ 
( { (/) }  X.  (tag  A  u. tag  C ) ) )  =  ( { 1o }  X. tag  B ) )
97, 8ax-mp 5 . . . . . . 7  |-  ( ( { 1o }  X. tag  B )  \  ( {
(/) }  X.  (tag  A  u. tag  C )
) )  =  ( { 1o }  X. tag  B )
10 bj-1ex 31455 . . . . . . . . . 10  |-  1o  e.  _V
1110snnz 4056 . . . . . . . . 9  |-  { 1o }  =/=  (/)
12 bj-tagn0 31484 . . . . . . . . 9  |- tag  B  =/=  (/)
1311, 12pm3.2i 456 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  /\ tag  B  =/=  (/) )
14 xpnz 5213 . . . . . . . 8  |-  ( ( { 1o }  =/=  (/) 
/\ tag  B  =/=  (/) )  <->  ( { 1o }  X. tag  B )  =/=  (/) )
1513, 14mpbi 211 . . . . . . 7  |-  ( { 1o }  X. tag  B
)  =/=  (/)
169, 15eqnetri 2666 . . . . . 6  |-  ( ( { 1o }  X. tag  B )  \  ( {
(/) }  X.  (tag  A  u. tag  C )
) )  =/=  (/)
172, 16eqnetrri 2667 . . . . 5  |-  ( ( { 1o }  X. tag  B )  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  =/=  (/)
18 0pss 3770 . . . . 5  |-  ( (/)  C.  ( ( { 1o }  X. tag  B )  \ 
( ( { (/) }  X. tag  A )  u.  ( { (/) }  X. tag  C ) ) )  <->  ( ( { 1o }  X. tag  B
)  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  =/=  (/) )
1917, 18mpbir 212 . . . 4  |-  (/)  C.  (
( { 1o }  X. tag  B )  \  (
( { (/) }  X. tag  A )  u.  ( {
(/) }  X. tag  C ) ) )
20 ssun2 3568 . . . . . . . 8  |-  ( {
(/) }  X. tag  C ) 
C_  ( ( {
(/) }  X. tag  A )  u.  ( { (/) }  X. tag  C ) )
21 sscon 3537 . . . . . . . 8  |-  ( ( { (/) }  X. tag  C
)  C_  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) )  ->  ( ( { 1o }  X. tag  B
)  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  C_  (
( { 1o }  X. tag  B )  \  ( { (/) }  X. tag  C
) ) )
2220, 21ax-mp 5 . . . . . . 7  |-  ( ( { 1o }  X. tag  B )  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  C_  (
( { 1o }  X. tag  B )  \  ( { (/) }  X. tag  C
) )
23 ssun2 3568 . . . . . . . 8  |-  ( { 1o }  X. tag  B
)  C_  ( ( { (/) }  X. tag  A
)  u.  ( { 1o }  X. tag  B
) )
24 ssdif 3538 . . . . . . . 8  |-  ( ( { 1o }  X. tag  B )  C_  ( ( { (/) }  X. tag  A
)  u.  ( { 1o }  X. tag  B
) )  ->  (
( { 1o }  X. tag  B )  \  ( { (/) }  X. tag  C
) )  C_  (
( ( { (/) }  X. tag  A )  u.  ( { 1o }  X. tag  B ) )  \ 
( { (/) }  X. tag  C ) ) )
2523, 24ax-mp 5 . . . . . . 7  |-  ( ( { 1o }  X. tag  B )  \  ( {
(/) }  X. tag  C ) )  C_  ( (
( { (/) }  X. tag  A )  u.  ( { 1o }  X. tag  B
) )  \  ( { (/) }  X. tag  C
) )
2622, 25sstri 3411 . . . . . 6  |-  ( ( { 1o }  X. tag  B )  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  C_  (
( ( { (/) }  X. tag  A )  u.  ( { 1o }  X. tag  B ) )  \ 
( { (/) }  X. tag  C ) )
27 df-bj-2upl 31516 . . . . . . . 8  |- (| A,  B|)  =  ((| A|)  u.  ( { 1o }  X. tag  B
) )
28 df-bj-1upl 31503 . . . . . . . . 9  |- (| A|)  =  ( { (/) }  X. tag  A
)
2928uneq1i 3554 . . . . . . . 8  |-  ((| A|)  u.  ( { 1o }  X. tag  B ) )  =  ( ( {
(/) }  X. tag  A )  u.  ( { 1o }  X. tag  B ) )
3027, 29eqtri 2445 . . . . . . 7  |- (| A,  B|)  =  ( ( {
(/) }  X. tag  A )  u.  ( { 1o }  X. tag  B ) )
3130difeq1i 3517 . . . . . 6  |-  ((| A,  B|)  \  ( {
(/) }  X. tag  C ) )  =  ( ( ( { (/) }  X. tag  A )  u.  ( { 1o }  X. tag  B
) )  \  ( { (/) }  X. tag  C
) )
3226, 31sseqtr4i 3435 . . . . 5  |-  ( ( { 1o }  X. tag  B )  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  C_  ((| A,  B|)  \  ( {
(/) }  X. tag  C ) )
33 df-bj-1upl 31503 . . . . . 6  |- (| C|)  =  ( { (/) }  X. tag  C
)
3433difeq2i 3518 . . . . 5  |-  ((| A,  B|)  \ (| C|) )  =  ((| A,  B|)  \ 
( { (/) }  X. tag  C ) )
3532, 34sseqtr4i 3435 . . . 4  |-  ( ( { 1o }  X. tag  B )  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  C_  ((| A,  B|)  \ (| C|) )
36 psssstr 3509 . . . 4  |-  ( (
(/)  C.  ( ( { 1o }  X. tag  B
)  \  ( ( { (/) }  X. tag  A
)  u.  ( {
(/) }  X. tag  C ) ) )  /\  (
( { 1o }  X. tag  B )  \  (
( { (/) }  X. tag  A )  u.  ( {
(/) }  X. tag  C ) ) )  C_  ((| A,  B|)  \ (| C|) )
)  ->  (/)  C.  ((| A,  B|)  \ (| C|) )
)
3719, 35, 36mp2an 676 . . 3  |-  (/)  C.  ((| A,  B|)  \ (| C|) )
38 0pss 3770 . . 3  |-  ( (/)  C.  ((| A,  B|)  \ (| C|) )  <->  ((| A,  B|)  \ (| C|) )  =/=  (/) )
3937, 38mpbi 211 . 2  |-  ((| A,  B|)  \ (| C|) )  =/=  (/)
40 difn0 3792 . 2  |-  ( ((| A,  B|)  \ (| C|) )  =/=  (/)  -> (| A,  B|)  =/= (| C|) )
4139, 40ax-mp 5 1  |- (| A,  B|)  =/= (| C|)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    =/= wne 2594    \ cdif 3371    u. cun 3372    i^i cin 3373    C_ wss 3374    C. wpss 3375   (/)c0 3699   {csn 3936    X. cxp 4789   1oc1o 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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