Theorem djurclALT 16520

Description: Shortening of djurcl 7311 using djucllem 16518. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
djurclALT	|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

Proof of Theorem djurclALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 6633	. . . . 5 |- 1o e. _V
2		df-inr 7307	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
3	1, 2	djucllem 16518	. . . 4 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( { 1o } X. B ) )
4	3	olcd 742	. . 3 |- ( C e. B -> ( ( (inr |` B ) ` C ) e. ( { (/) } X. A ) \/ ( (inr |` B ) ` C ) e. ( { 1o } X. B ) ) )
5		elun 3350	. . 3 |- ( ( (inr |` B ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) <-> ( ( (inr |` B ) ` C ) e. ( { (/) } X. A ) \/ ( (inr |` B ) ` C ) e. ( { 1o } X. B ) ) )
6	4, 5	sylibr 134	. 2 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) )
7		df-dju 7297	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
8	6, 7	eleqtrrdi 2325	1 |- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

Syntax hints:   -> wi 4   \/ wo 716   e. wcel 2202   u. cun 3199   (/)c0 3496   {csn 3673   X. cxp 4729   |` cres 4733   ` cfv 5333   1oc1o 6618   ⊔ cdju 7296  inrcinr 7305

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-1o 6625  df-dju 7297  df-inr 7307

This theorem is referenced by: (None)