Theorem djulclb 7385

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 7381	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
2		1n0 6695	. . . . . . . . . 10 ⊢ 1o ≠ ∅
3	2	necomi 2505	. . . . . . . . 9 ⊢ ∅ ≠ 1o
4		0ex 4255	. . . . . . . . . 10 ⊢ ∅ ∈ V
5	4	elsn 3721	. . . . . . . . 9 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
6	3, 5	nemtbir 2509	. . . . . . . 8 ⊢ ¬ ∅ ∈ {1o}
7	6	intnanr 942	. . . . . . 7 ⊢ ¬ (∅ ∈ {1o} ∧ 𝐶 ∈ 𝐵)
8		opelxp 4799	. . . . . . 7 ⊢ (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ↔ (∅ ∈ {1o} ∧ 𝐶 ∈ 𝐵))
9	7, 8	mtbir 682	. . . . . 6 ⊢ ¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)
10		elex 2833	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
11		opexg 4363	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ V ∧ 𝐶 ∈ 𝑉) → ⟨∅, 𝐶⟩ ∈ V)
12	4, 11	mpan 428	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑉 → ⟨∅, 𝐶⟩ ∈ V)
13		opeq2 3900	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
14		df-inl 7377	. . . . . . . . . . . . 13 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	13, 14	fvmptg 5775	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ V) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
16	10, 12, 15	syl2anc 415	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝑉 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
17	16	adantr 276	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
18		df-dju 7368	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
19	18	eleq2i 2305	. . . . . . . . . . . 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 2316	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
23		elun 3370	. . . . . . . . 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 741	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ∨ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
26	25	ord 736	. . . . . 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 4799	. . . . 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 720   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218  ∅c0 3520  {csn 3705  ⟨cop 3708   × cxp 4767  ‘cfv 5372  1_oc1o 6670   ⊔ cdju 7367  inlcinl 7375

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-1o 6677  df-dju 7368  df-inl 7377

This theorem is referenced by:  exmidfodomrlemr  7544