Theorem eldju1st 7048

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 7047	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		ssel2 3142	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
3		xp1st 6144	. . 3 ⊢ (𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → (1st ‘𝑋) ∈ {∅, 1o})
4		elpri 3606	. . 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 422	1 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 703   = wceq 1348   ∈ wcel 2141   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {cpr 3584   × cxp 4609  ‘cfv 5198  1^st c1st 6117  1_oc1o 6388   ⊔ cdju 7014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025

This theorem is referenced by:  updjudhf  7056  subctctexmid  14034