Theorem djuss 9813

Description: A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.)

Assertion

Ref	Expression
djuss	⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))

Proof of Theorem djuss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		djur 9812	. . 3 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)))
2		simpr 484	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3		df-inl 9795	. . . . . . . . 9 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4		opeq2 4826	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5		elex 3457	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ V)
6		opex 5404	. . . . . . . . . 10 ⊢ ⟨∅, 𝑦⟩ ∈ V
7	6	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ⟨∅, 𝑦⟩ ∈ V)
8	3, 4, 5, 7	fvmptd3 6952	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
10	2, 9	eqtrd 2766	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
11		elun1 4132	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (𝐴 ∪ 𝐵))
12		0ex 5245	. . . . . . . . . 10 ⊢ ∅ ∈ V
13	12	prid1 4715	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
14	11, 13	jctil 519	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
15	14	adantr 480	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
16		opelxp 5652	. . . . . . 7 ⊢ (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
17	15, 16	sylibr 234	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
18	10, 17	eqeltrd 2831	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
19	18	rexlimiva 3125	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
20		simpr 484	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
21		df-inr 9796	. . . . . . . . 9 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
22		opeq2 4826	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
23		elex 3457	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ V)
24		opex 5404	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ⟨1o, 𝑦⟩ ∈ V)
26	21, 22, 23, 25	fvmptd3 6952	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
27	26	adantr 480	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
28	20, 27	eqtrd 2766	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
29		elun2 4133	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴 ∪ 𝐵))
31		1oex 8395	. . . . . . . . 9 ⊢ 1o ∈ V
32	31	prid2 4716	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
33	30, 32	jctil 519	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
34		opelxp 5652	. . . . . . 7 ⊢ (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
35	33, 34	sylibr 234	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
36	28, 35	eqeltrd 2831	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
37	36	rexlimiva 3125	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
38	19, 37	jaoi 857	. . 3 ⊢ ((∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
39	1, 38	syl 17	. 2 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
40	39	ssriv 3938	1 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  {cpr 4578  ⟨cop 4582   × cxp 5614  ‘cfv 6481  1_oc1o 8378   ⊔ cdju 9791  inlcinl 9792  inrcinr 9793

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-suc 6312  df-iota 6437  df-fun 6483  df-fv 6489  df-1st 7921  df-2nd 7922  df-1o 8385  df-dju 9794  df-inl 9795  df-inr 9796

This theorem is referenced by:  djuunxp  9814  djuexALT  9815  eldju1st  9816