Theorem eldju1st 7275

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

Assertion

Ref	Expression
eldju1st	⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))

Proof of Theorem eldju1st

Step	Hyp	Ref	Expression
1		djuss 7274	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		ssel2 3221	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
3		xp1st 6333	. . 3 ⊢ (𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → (1st ‘𝑋) ∈ {∅, 1o})
4		elpri 3693	. . 3 ⊢ ((1st ‘𝑋) ∈ {∅, 1o} → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
5	2, 3, 4	3syl 17	. 2 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
6	1, 5	mpan 424	1 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 715   = wceq 1397   ∈ wcel 2201   ∪ cun 3197   ⊆ wss 3199  ∅c0 3493  {cpr 3671   × cxp 4725  ‘cfv 5328  1^st c1st 6306  1_oc1o 6580   ⊔ cdju 7241

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-1st 6308  df-2nd 6309  df-1o 6587  df-dju 7242  df-inl 7251  df-inr 7252

This theorem is referenced by:  updjudhf  7283  subctctexmid  16661