Theorem djulclb 7359

Description: Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)

Assertion

Ref	Expression
djulclb	⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ 𝐴 ↔ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)))

Proof of Theorem djulclb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djulcl 7355	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
2		1n0 6678	. . . . . . . . . 10 ⊢ 1o ≠ ∅
3	2	necomi 2499	. . . . . . . . 9 ⊢ ∅ ≠ 1o
4		0ex 4242	. . . . . . . . . 10 ⊢ ∅ ∈ V
5	4	elsn 3710	. . . . . . . . 9 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
6	3, 5	nemtbir 2503	. . . . . . . 8 ⊢ ¬ ∅ ∈ {1o}
7	6	intnanr 938	. . . . . . 7 ⊢ ¬ (∅ ∈ {1o} ∧ 𝐶 ∈ 𝐵)
8		opelxp 4784	. . . . . . 7 ⊢ (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ↔ (∅ ∈ {1o} ∧ 𝐶 ∈ 𝐵))
9	7, 8	mtbir 678	. . . . . 6 ⊢ ¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)
10		elex 2827	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
11		opexg 4349	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ V ∧ 𝐶 ∈ 𝑉) → ⟨∅, 𝐶⟩ ∈ V)
12	4, 11	mpan 424	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑉 → ⟨∅, 𝐶⟩ ∈ V)
13		opeq2 3889	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
14		df-inl 7351	. . . . . . . . . . . . 13 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	13, 14	fvmptg 5758	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ V) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
16	10, 12, 15	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝑉 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
17	16	adantr 276	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
18		df-dju 7342	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
19	18	eleq2i 2301	. . . . . . . . . . . 12 ⊢ ((inl‘𝐶) ∈ (𝐴 ⊔ 𝐵) ↔ (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
20	19	biimpi 120	. . . . . . . . . . 11 ⊢ ((inl‘𝐶) ∈ (𝐴 ⊔ 𝐵) → (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
21	20	adantl 277	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
22	17, 21	eqeltrrd 2312	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
23		elun 3364	. . . . . . . . 9 ⊢ (⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ∨ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)))
24	22, 23	sylib 122	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ∨ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)))
25	24	orcomd 737	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ∨ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
26	25	ord 732	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
27	9, 26	mpi 15	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
28		opelxp 4784	. . . . 5 ⊢ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ↔ (∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴))
29	27, 28	sylib 122	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴))
30	29	simprd 114	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → 𝐶 ∈ 𝐴)
31	30	ex 115	. 2 ⊢ (𝐶 ∈ 𝑉 → ((inl‘𝐶) ∈ (𝐴 ⊔ 𝐵) → 𝐶 ∈ 𝐴))
32	1, 31	impbid2 143	1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ 𝐴 ↔ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∪ cun 3212  ∅c0 3512  {csn 3694  ⟨cop 3697   × cxp 4752  ‘cfv 5357  1_oc1o 6653   ⊔ cdju 7341  inlcinl 7349

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-1o 6660  df-dju 7342  df-inl 7351

This theorem is referenced by:  exmidfodomrlemr  7518