Theorem djuss 6963

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 6962	. . 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 6940	. . . . . . . . 9 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4		opeq2 3714	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5		elex 2700	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ V)
6		0ex 4063	. . . . . . . . . . 11 ⊢ ∅ ∈ V
7		vex 2692	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	6, 7	opex 4159	. . . . . . . . . 10 ⊢ ⟨∅, 𝑦⟩ ∈ V
9	8	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → ⟨∅, 𝑦⟩ ∈ V)
10	3, 4, 5, 9	fvmptd3 5522	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
11	10	adantr 274	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
12	2, 11	eqtrd 2173	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
13		elun1 3248	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (𝐴 ∪ 𝐵))
14	6	prid1 3637	. . . . . . . . 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 4577	. . . . . . 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 2217	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
20	19	rexlimiva 2547	. . . 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 6941	. . . . . . . . 9 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
23		opeq2 3714	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
24		elex 2700	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ V)
25		1oex 6329	. . . . . . . . . . 11 ⊢ 1o ∈ V
26	25, 7	opex 4159	. . . . . . . . . 10 ⊢ ⟨1o, 𝑦⟩ ∈ V
27	26	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ⟨1o, 𝑦⟩ ∈ V)
28	22, 23, 24, 27	fvmptd3 5522	. . . . . . . 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 2173	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
31		elun2 3249	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐴 ∪ 𝐵))
32	31	adantr 274	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴 ∪ 𝐵))
33	25	prid2 3638	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
34	32, 33	jctil 310	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
35		opelxp 4577	. . . . . . 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 2217	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
38	37	rexlimiva 2547	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)))
39	20, 38	jaoi 706	. . 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 3106	1 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  Vcvv 2689   ∪ cun 3074   ⊆ wss 3076  ∅c0 3368  {cpr 3533  ⟨cop 3535   × cxp 4545  ‘cfv 5131  1_oc1o 6314   ⊔ cdju 6930  inlcinl 6938  inrcinr 6939

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941

This theorem is referenced by:  eldju1st  6964