Theorem eldju1st 9352

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 9349	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		ssel2 3962	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
3		xp1st 7721	. . 3 ⊢ (𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → (1st ‘𝑋) ∈ {∅, 1o})
4		elpri 4589	. . 3 ⊢ ((1st ‘𝑋) ∈ {∅, 1o} → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
5	2, 3, 4	3syl 18	. 2 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
6	1, 5	mpan 688	1 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  {cpr 4569   × cxp 5553  ‘cfv 6355  1^st c1st 7687  1_oc1o 8095   ⊔ cdju 9327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fv 6363  df-1st 7689  df-2nd 7690  df-1o 8102  df-dju 9330  df-inl 9331  df-inr 9332

This theorem is referenced by: (None)