Theorem unidif0 4201

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 3457	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → ¬ 𝑦 = ∅)
2	1	pm4.71i 391	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅))
3	2	anbi1i 458	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ((𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦 ∈ 𝐴))
4		an32 562	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 = ∅) ↔ ((𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦 ∈ 𝐴))
5		anass 401	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 = ∅) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
6	3, 4, 5	3bitr2ri 209	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
7	6	exbii 1619	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 3843	. . . 4 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})))
9		eldif 3166	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10		velsn 3640	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	10	notbii 669	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {∅} ↔ ¬ 𝑦 = ∅)
12	11	anbi2i 457	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅))
13	9, 12	bitri 184	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅))
14	13	anbi2i 457	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
15	14	exbii 1619	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
16	8, 15	bitri 184	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
17		eluni 3843	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
18	7, 16, 17	3bitr4i 212	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ 𝑥 ∈ ∪ 𝐴)
19	18	eqriv 2193	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167   ∖ cdif 3154  ∅c0 3451  {csn 3623  ∪ cuni 3840

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3452  df-sn 3629  df-uni 3841

This theorem is referenced by: (None)