Theorem unidif0 4153

Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Assertion

Ref	Expression
unidif0	⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Proof of Theorem unidif0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 3420	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → ¬ 𝑦 = ∅)
2	1	pm4.71i 389	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅))
3	2	anbi1i 455	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ((𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦 ∈ 𝐴))
4		an32 557	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 = ∅) ↔ ((𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦 ∈ 𝐴))
5		anass 399	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 = ∅) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
6	3, 4, 5	3bitr2ri 208	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
7	6	exbii 1598	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 3799	. . . 4 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})))
9		eldif 3130	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10		velsn 3600	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	10	notbii 663	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {∅} ↔ ¬ 𝑦 = ∅)
12	11	anbi2i 454	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅))
13	9, 12	bitri 183	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅))
14	13	anbi2i 454	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
15	14	exbii 1598	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
16	8, 15	bitri 183	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
17		eluni 3799	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
18	7, 16, 17	3bitr4i 211	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ 𝑥 ∈ ∪ 𝐴)
19	18	eqriv 2167	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ∖ cdif 3118  ∅c0 3414  {csn 3583  ∪ cuni 3796

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3589  df-uni 3797

This theorem is referenced by: (None)