Theorem iununir 3729

Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
iununir	⊢ ((A ∪ ∪ B) = ∪ x ∈ B (A ∪ x) → (B = ∅ → A = ∅))

Distinct variable groups:   x,A   x,B

Proof of Theorem iununir

Step	Hyp	Ref	Expression
1		unieq 3580	. . . . . 6 ⊢ (B = ∅ → ∪ B = ∪ ∅)
2		uni0 3598	. . . . . 6 ⊢ ∪ ∅ = ∅
3	1, 2	syl6eq 2085	. . . . 5 ⊢ (B = ∅ → ∪ B = ∅)
4	3	uneq2d 3091	. . . 4 ⊢ (B = ∅ → (A ∪ ∪ B) = (A ∪ ∅))
5		un0 3245	. . . 4 ⊢ (A ∪ ∅) = A
6	4, 5	syl6eq 2085	. . 3 ⊢ (B = ∅ → (A ∪ ∪ B) = A)
7		iuneq1 3661	. . . 4 ⊢ (B = ∅ → ∪ x ∈ B (A ∪ x) = ∪ x ∈ ∅ (A ∪ x))
8		0iun 3705	. . . 4 ⊢ ∪ x ∈ ∅ (A ∪ x) = ∅
9	7, 8	syl6eq 2085	. . 3 ⊢ (B = ∅ → ∪ x ∈ B (A ∪ x) = ∅)
10	6, 9	eqeq12d 2051	. 2 ⊢ (B = ∅ → ((A ∪ ∪ B) = ∪ x ∈ B (A ∪ x) ↔ A = ∅))
11	10	biimpcd 148	1 ⊢ ((A ∪ ∪ B) = ∪ x ∈ B (A ∪ x) → (B = ∅ → A = ∅))

Syntax hints:   → wi 4   = wceq 1242   ∪ cun 2909  ∅c0 3218  ∪ cuni 3571  ∪ ciun 3648

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572  df-iun 3650

This theorem is referenced by: (None)