Theorem djuss 6955

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 6954	. . 3 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)))
2		simpr 109	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3		df-inl 6932	. . . . . . . . 9 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4		opeq2 3706	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5		elex 2697	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ V)
6		0ex 4055	. . . . . . . . . . 11 ⊢ ∅ ∈ V
7		vex 2689	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	6, 7	opex 4151	. . . . . . . . . 10 ⊢ ⟨∅, 𝑦⟩ ∈ V
9	8	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ⟨∅, 𝑦⟩ ∈ V)
10	3, 4, 5, 9	fvmptd3 5514	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
11	10	adantr 274	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
12	2, 11	eqtrd 2172	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
13		elun1 3243	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (𝐴 ∪ 𝐵))
14	6	prid1 3629	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
15	13, 14	jctil 310	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
16	15	adantr 274	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
17		opelxp 4569	. . . . . . 7 ⊢ (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
18	16, 17	sylibr 133	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
19	12, 18	eqeltrd 2216	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
20	19	rexlimiva 2544	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
21		simpr 109	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
22		df-inr 6933	. . . . . . . . 9 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
23		opeq2 3706	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
24		elex 2697	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ V)
25		1oex 6321	. . . . . . . . . . 11 ⊢ 1o ∈ V
26	25, 7	opex 4151	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
27	26	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ⟨1o, 𝑦⟩ ∈ V)
28	22, 23, 24, 27	fvmptd3 5514	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
29	28	adantr 274	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
30	21, 29	eqtrd 2172	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
31		elun2 3244	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
32	31	adantr 274	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴 ∪ 𝐵))
33	25	prid2 3630	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
34	32, 33	jctil 310	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
35		opelxp 4569	. . . . . . 7 ⊢ (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
36	34, 35	sylibr 133	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
37	30, 36	eqeltrd 2216	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
38	37	rexlimiva 2544	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
39	20, 38	jaoi 705	. . 3 ⊢ ((∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
40	1, 39	sylbi 120	. 2 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
41	40	ssriv 3101	1 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))

Syntax hints:   ∧ wa 103   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686   ∪ cun 3069   ⊆ wss 3071  ∅c0 3363  {cpr 3528  ⟨cop 3530   × cxp 4537  ‘cfv 5123  1_oc1o 6306   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933

This theorem is referenced by:  eldju1st  6956