Theorem eldju1st 9847

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 9844	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		ssel2 3930	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
3		xp1st 7975	. . 3 ⊢ (𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → (1st ‘𝑋) ∈ {∅, 1o})
4		elpri 4606	. . 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 691	1 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  {cpr 4584   × cxp 5630  ‘cfv 6500  1^st c1st 7941  1_oc1o 8400   ⊔ cdju 9822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-suc 6331  df-iota 6456  df-fun 6502  df-fv 6508  df-1st 7943  df-2nd 7944  df-1o 8407  df-dju 9825  df-inl 9826  df-inr 9827

This theorem is referenced by: (None)