Theorem eldju1st 7270

Description: The first component of an element of a disjoint union is either (/) or 1o. (Contributed by AV, 26-Jun-2022.)

Assertion

Ref	Expression
eldju1st	|- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )

Proof of Theorem eldju1st

Step	Hyp	Ref	Expression
1		djuss 7269	. 2 |- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
2		ssel2 3222	. . 3 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> X e. ( { (/) , 1o } X. ( A u. B ) ) )
3		xp1st 6328	. . 3 |- ( X e. ( { (/) , 1o } X. ( A u. B ) ) -> ( 1st ` X ) e. { (/) , 1o } )
4		elpri 3692	. . 3 |- ( ( 1st ` X ) e. { (/) , 1o } -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )
5	2, 3, 4	3syl 17	. 2 |- ( ( ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) ) /\ X e. ( A ⊔ B ) ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )
6	1, 5	mpan 424	1 |- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   u. cun 3198   C_ wss 3200   (/)c0 3494   {cpr 3670   X. cxp 4723   ` cfv 5326   1stc1st 6301   1oc1o 6575   ⊔ cdju 7236

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6303  df-2nd 6304  df-1o 6582  df-dju 7237  df-inl 7246  df-inr 7247

This theorem is referenced by:  updjudhf  7278  subctctexmid  16622