Theorem unidif0OLD 5319

Description: Obsolete version of unidif0 5318 as of 25-Apr-2026. (Contributed by NM, 22-Mar-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unidif0OLD	⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Proof of Theorem unidif0OLD

Step	Hyp	Ref	Expression
1		uniun 4890	. . . 4 ⊢ ∪ ((𝐴 ∖ {∅}) ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
2		undif1 4432	. . . . . 6 ⊢ ((𝐴 ∖ {∅}) ∪ {∅}) = (𝐴 ∪ {∅})
3		uncom 4113	. . . . . 6 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
4	2, 3	eqtr2i 2788	. . . . 5 ⊢ ({∅} ∪ 𝐴) = ((𝐴 ∖ {∅}) ∪ {∅})
5	4	unieqi 4879	. . . 4 ⊢ ∪ ({∅} ∪ 𝐴) = ∪ ((𝐴 ∖ {∅}) ∪ {∅})
6		0ex 5259	. . . . . . 7 ⊢ ∅ ∈ V
7	6	unisn 4886	. . . . . 6 ⊢ ∪ {∅} = ∅
8	7	uneq2i 4120	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∅)
9		un0 4350	. . . . 5 ⊢ (∪ (𝐴 ∖ {∅}) ∪ ∅) = ∪ (𝐴 ∖ {∅})
10	8, 9	eqtr2i 2788	. . . 4 ⊢ ∪ (𝐴 ∖ {∅}) = (∪ (𝐴 ∖ {∅}) ∪ ∪ {∅})
11	1, 5, 10	3eqtr4ri 2798	. . 3 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ ({∅} ∪ 𝐴)
12		uniun 4890	. . 3 ⊢ ∪ ({∅} ∪ 𝐴) = (∪ {∅} ∪ ∪ 𝐴)
13	7	uneq1i 4119	. . 3 ⊢ (∪ {∅} ∪ ∪ 𝐴) = (∅ ∪ ∪ 𝐴)
14	11, 12, 13	3eqtri 2791	. 2 ⊢ ∪ (𝐴 ∖ {∅}) = (∅ ∪ ∪ 𝐴)
15		uncom 4113	. 2 ⊢ (∅ ∪ ∪ 𝐴) = (∪ 𝐴 ∪ ∅)
16		un0 4350	. 2 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
17	14, 15, 16	3eqtri 2791	1 ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

Syntax hints:   = wceq 1562   ∖ cdif 3903   ∪ cun 3904  ∅c0 4287  {csn 4584  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is referenced by: (None)