Theorem djulclb 7157

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 7153	. 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
2		1n0 6518	. . . . . . . . . 10 ⊢ 1o ≠ ∅
3	2	necomi 2461	. . . . . . . . 9 ⊢ ∅ ≠ 1o
4		0ex 4171	. . . . . . . . . 10 ⊢ ∅ ∈ V
5	4	elsn 3649	. . . . . . . . 9 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
6	3, 5	nemtbir 2465	. . . . . . . 8 ⊢ ¬ ∅ ∈ {1o}
7	6	intnanr 932	. . . . . . 7 ⊢ ¬ (∅ ∈ {1o} ∧ 𝐶 ∈ 𝐵)
8		opelxp 4705	. . . . . . 7 ⊢ (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ↔ (∅ ∈ {1o} ∧ 𝐶 ∈ 𝐵))
9	7, 8	mtbir 673	. . . . . 6 ⊢ ¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)
10		elex 2783	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
11		opexg 4272	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ V ∧ 𝐶 ∈ 𝑉) → ⟨∅, 𝐶⟩ ∈ V)
12	4, 11	mpan 424	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑉 → ⟨∅, 𝐶⟩ ∈ V)
13		opeq2 3820	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
14		df-inl 7149	. . . . . . . . . . . . 13 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	13, 14	fvmptg 5655	. . . . . . . . . . . 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 7140	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
19	18	eleq2i 2272	. . . . . . . . . . . 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 2283	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
23		elun 3314	. . . . . . . . 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 731	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) → (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ∨ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
26	25	ord 726	. . . . . 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 4705	. . . . 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 710   = wceq 1373   ∈ wcel 2176  Vcvv 2772   ∪ cun 3164  ∅c0 3460  {csn 3633  ⟨cop 3636   × cxp 4673  ‘cfv 5271  1_oc1o 6495   ⊔ cdju 7139  inlcinl 7147

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-1o 6502  df-dju 7140  df-inl 7149

This theorem is referenced by:  exmidfodomrlemr  7310