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Theorem endjusym 7097
Description: Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
Assertion
Ref Expression
endjusym  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  ~~  ( B A )
)

Proof of Theorem endjusym
StepHypRef Expression
1 djulf1o 7059 . . . . . . . . 9  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
2 f1of1 5462 . . . . . . . . 9  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  -> inl : _V -1-1-> ( {
(/) }  X.  _V )
)
31, 2ax-mp 5 . . . . . . . 8  |- inl : _V -1-1-> ( { (/) }  X.  _V )
4 ssv 3179 . . . . . . . 8  |-  A  C_  _V
5 f1ores 5478 . . . . . . . 8  |-  ( (inl : _V -1-1-> ( {
(/) }  X.  _V )  /\  A  C_  _V )  ->  (inl  |`  A ) : A -1-1-onto-> (inl " A ) )
63, 4, 5mp2an 426 . . . . . . 7  |-  (inl  |`  A ) : A -1-1-onto-> (inl " A )
7 f1oeng 6759 . . . . . . 7  |-  ( ( A  e.  V  /\  (inl  |`  A ) : A -1-1-onto-> (inl " A ) )  ->  A  ~~  (inl " A ) )
86, 7mpan2 425 . . . . . 6  |-  ( A  e.  V  ->  A  ~~  (inl " A ) )
98ensymd 6785 . . . . 5  |-  ( A  e.  V  ->  (inl " A )  ~~  A
)
10 djurf1o 7060 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
11 f1of1 5462 . . . . . . . 8  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  -> inr : _V -1-1-> ( { 1o }  X.  _V ) )
1210, 11ax-mp 5 . . . . . . 7  |- inr : _V -1-1-> ( { 1o }  X.  _V )
13 f1ores 5478 . . . . . . 7  |-  ( (inr : _V -1-1-> ( { 1o }  X.  _V )  /\  A  C_  _V )  ->  (inr  |`  A ) : A -1-1-onto-> (inr " A ) )
1412, 4, 13mp2an 426 . . . . . 6  |-  (inr  |`  A ) : A -1-1-onto-> (inr " A )
15 f1oeng 6759 . . . . . 6  |-  ( ( A  e.  V  /\  (inr  |`  A ) : A -1-1-onto-> (inr " A ) )  ->  A  ~~  (inr " A ) )
1614, 15mpan2 425 . . . . 5  |-  ( A  e.  V  ->  A  ~~  (inr " A ) )
17 entr 6786 . . . . 5  |-  ( ( (inl " A ) 
~~  A  /\  A  ~~  (inr " A ) )  ->  (inl " A
)  ~~  (inr " A
) )
189, 16, 17syl2anc 411 . . . 4  |-  ( A  e.  V  ->  (inl " A )  ~~  (inr " A ) )
1918adantr 276 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inl " A ) 
~~  (inr " A
) )
20 ssv 3179 . . . . . . . 8  |-  B  C_  _V
21 f1ores 5478 . . . . . . . 8  |-  ( (inr : _V -1-1-> ( { 1o }  X.  _V )  /\  B  C_  _V )  ->  (inr  |`  B ) : B -1-1-onto-> (inr " B ) )
2212, 20, 21mp2an 426 . . . . . . 7  |-  (inr  |`  B ) : B -1-1-onto-> (inr " B )
23 f1oeng 6759 . . . . . . 7  |-  ( ( B  e.  W  /\  (inr  |`  B ) : B -1-1-onto-> (inr " B ) )  ->  B  ~~  (inr " B ) )
2422, 23mpan2 425 . . . . . 6  |-  ( B  e.  W  ->  B  ~~  (inr " B ) )
2524adantl 277 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  ~~  (inr " B ) )
2625ensymd 6785 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inr " B ) 
~~  B )
27 f1ores 5478 . . . . . . 7  |-  ( (inl : _V -1-1-> ( {
(/) }  X.  _V )  /\  B  C_  _V )  ->  (inl  |`  B ) : B -1-1-onto-> (inl " B ) )
283, 20, 27mp2an 426 . . . . . 6  |-  (inl  |`  B ) : B -1-1-onto-> (inl " B )
29 f1oeng 6759 . . . . . 6  |-  ( ( B  e.  W  /\  (inl  |`  B ) : B -1-1-onto-> (inl " B ) )  ->  B  ~~  (inl " B ) )
3028, 29mpan2 425 . . . . 5  |-  ( B  e.  W  ->  B  ~~  (inl " B ) )
3130adantl 277 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  ~~  (inl " B ) )
32 entr 6786 . . . 4  |-  ( ( (inr " B ) 
~~  B  /\  B  ~~  (inl " B ) )  ->  (inr " B
)  ~~  (inl " B
) )
3326, 31, 32syl2anc 411 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inr " B ) 
~~  (inl " B
) )
34 djuin 7065 . . . 4  |-  ( (inl " A )  i^i  (inr " B ) )  =  (/)
3534a1i 9 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (inl " A
)  i^i  (inr " B
) )  =  (/) )
36 incom 3329 . . . . 5  |-  ( (inl " B )  i^i  (inr " A ) )  =  ( (inr " A
)  i^i  (inl " B
) )
37 djuin 7065 . . . . 5  |-  ( (inl " B )  i^i  (inr " A ) )  =  (/)
3836, 37eqtr3i 2200 . . . 4  |-  ( (inr " A )  i^i  (inl " B ) )  =  (/)
3938a1i 9 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (inr " A
)  i^i  (inl " B
) )  =  (/) )
40 unen 6818 . . 3  |-  ( ( ( (inl " A
)  ~~  (inr " A
)  /\  (inr " B
)  ~~  (inl " B
) )  /\  (
( (inl " A
)  i^i  (inr " B
) )  =  (/)  /\  ( (inr " A
)  i^i  (inl " B
) )  =  (/) ) )  ->  (
(inl " A )  u.  (inr " B ) )  ~~  ( (inr " A )  u.  (inl " B ) ) )
4119, 33, 35, 39, 40syl22anc 1239 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (inl " A
)  u.  (inr " B ) )  ~~  ( (inr " A )  u.  (inl " B
) ) )
42 djuun 7068 . 2  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
43 uncom 3281 . . 3  |-  ( (inr " A )  u.  (inl " B ) )  =  ( (inl " B
)  u.  (inr " A ) )
44 djuun 7068 . . 3  |-  ( (inl " B )  u.  (inr " A ) )  =  ( B A )
4543, 44eqtri 2198 . 2  |-  ( (inr " A )  u.  (inl " B ) )  =  ( B A )
4641, 42, 453brtr3g 4038 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  ~~  ( B A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2739    u. cun 3129    i^i cin 3130    C_ wss 3131   (/)c0 3424   {csn 3594   class class class wbr 4005    X. cxp 4626    |` cres 4630   "cima 4631   -1-1->wf1 5215   -1-1-onto->wf1o 5217   1oc1o 6412    ~~ cen 6740   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-er 6537  df-en 6743  df-dju 7039  df-inl 7048  df-inr 7049
This theorem is referenced by:  sbthom  14813
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