Theorem unidif0 3920

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 3229	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → ¬ 𝑦 = ∅)
2	1	pm4.71i 371	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅))
3	2	anbi1i 431	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ((𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦 ∈ 𝐴))
4		an32 496	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 = ∅) ↔ ((𝑥 ∈ 𝑦 ∧ ¬ 𝑦 = ∅) ∧ 𝑦 ∈ 𝐴))
5		anass 381	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 = ∅) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
6	3, 4, 5	3bitr2ri 198	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
7	6	exbii 1496	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 3583	. . . 4 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})))
9		eldif 2927	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ {∅}))
10		velsn 3392	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	10	notbii 594	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {∅} ↔ ¬ 𝑦 = ∅)
12	11	anbi2i 430	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅))
13	9, 12	bitri 173	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ {∅}) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅))
14	13	anbi2i 430	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
15	14	exbii 1496	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∖ {∅})) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
16	8, 15	bitri 173	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = ∅)))
17		eluni 3583	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
18	7, 16, 17	3bitr4i 201	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∖ {∅}) ↔ 𝑥 ∈ ∪ 𝐴)
19	18	eqriv 2037	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ∖ cdif 2914  ∅c0 3224  {csn 3375  ∪ cuni 3580

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-sn 3381  df-uni 3581

This theorem is referenced by: (None)