Theorem djuss 7069

Description: A disjoint union is a subset 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 7068	. . 3 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)))
2		simpr 110	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3		df-inl 7046	. . . . . . . . 9 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4		opeq2 3780	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5		elex 2749	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ V)
6		0ex 4131	. . . . . . . . . . 11 ⊢ ∅ ∈ V
7		vex 2741	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	6, 7	opex 4230	. . . . . . . . . 10 ⊢ ⟨∅, 𝑦⟩ ∈ V
9	8	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ⟨∅, 𝑦⟩ ∈ V)
10	3, 4, 5, 9	fvmptd3 5610	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
11	10	adantr 276	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
12	2, 11	eqtrd 2210	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
13		elun1 3303	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (𝐴 ∪ 𝐵))
14	6	prid1 3699	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
15	13, 14	jctil 312	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
16	15	adantr 276	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
17		opelxp 4657	. . . . . . 7 ⊢ (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
18	16, 17	sylibr 134	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
19	12, 18	eqeltrd 2254	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
20	19	rexlimiva 2589	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
21		simpr 110	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
22		df-inr 7047	. . . . . . . . 9 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
23		opeq2 3780	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
24		elex 2749	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ V)
25		1oex 6425	. . . . . . . . . . 11 ⊢ 1o ∈ V
26	25, 7	opex 4230	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
27	26	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ⟨1o, 𝑦⟩ ∈ V)
28	22, 23, 24, 27	fvmptd3 5610	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
29	28	adantr 276	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
30	21, 29	eqtrd 2210	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
31		elun2 3304	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
32	31	adantr 276	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴 ∪ 𝐵))
33	25	prid2 3700	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
34	32, 33	jctil 312	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
35		opelxp 4657	. . . . . . 7 ⊢ (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
36	34, 35	sylibr 134	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
37	30, 36	eqeltrd 2254	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
38	37	rexlimiva 2589	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
39	20, 38	jaoi 716	. . 3 ⊢ ((∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
40	1, 39	sylbi 121	. 2 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
41	40	ssriv 3160	1 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))

Syntax hints:   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  Vcvv 2738   ∪ cun 3128   ⊆ wss 3130  ∅c0 3423  {cpr 3594  ⟨cop 3596   × cxp 4625  ‘cfv 5217  1_oc1o 6410   ⊔ cdju 7036  inlcinl 7044  inrcinr 7045

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047

This theorem is referenced by:  eldju1st  7070