Theorem unisnALT 44142

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. The User manually input on a mmj2 Proof Worksheet, without labels, all steps of unisnALT 44142 except 1, 11, 15, 21, and 30. With execution of the mmj2 unification command, mmj2 could find labels for all steps except for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15, 21, and 30). mmj2 could not find reference theorems for those five steps because the hypothesis field of each of these steps was empty and none of those steps unifies with a theorem in set.mm. Each of these five steps is a semantic variation of a theorem in set.mm and is 2-step provable. mmj2 does not have the ability to automatically generate the semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis deduction whose hypothesis is a theorem in set.mm which unifies with the theorem in the Proof Worksheet. The stepprover.c program, which invokes mmj2, has this capability. stepprover.c automatically generated steps 1, 11, 15, 21, and 30, labeled all steps, and generated the RPN proof of unisnALT 44142. Roughly speaking, stepprover.c added to the Proof Worksheet a labeled duplicate step of each non-unifying theorem for each label in a text file, labels.txt, containing a list of labels provided by the User. Upon mmj2 unification, stepprover.c identified a label for each of the five theorems which 2-step proves it. For unisnALT 44142, the label list is a list of all 1-hypothesis propositional calculus deductions in set.mm. stepproverp.c is the same as stepprover.c except that it intermittently pauses during execution, allowing the User to observe the changes to a text file caused by the execution of particular statements of the program. (Contributed by Alan Sare, 19-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
unisnALT.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
unisnALT	⊢ ∪ {𝐴} = 𝐴

Proof of Theorem unisnALT

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

Step	Hyp	Ref	Expression
1		eluni 4902	. . . . . 6 ⊢ (𝑥 ∈ ∪ {𝐴} ↔ ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}))
2	1	biimpi 215	. . . . 5 ⊢ (𝑥 ∈ ∪ {𝐴} → ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}))
3		id 22	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → (𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}))
4		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞)
6		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
7	3, 6	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8		elsni 4637	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
9	7, 8	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10		eleq2 2814	. . . . . . . . 9 ⊢ (𝑞 = 𝐴 → (𝑥 ∈ 𝑞 ↔ 𝑥 ∈ 𝐴))
11	10	biimpac 478	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 = 𝐴) → 𝑥 ∈ 𝐴)
12	5, 9, 11	syl2anc 583	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
13	12	ax-gen 1789	. . . . . 6 ⊢ ∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
14		19.23v 1937	. . . . . . 7 ⊢ (∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) ↔ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴))
15	14	biimpi 215	. . . . . 6 ⊢ (∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) → (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴))
16	13, 15	ax-mp 5	. . . . 5 ⊢ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
17		pm3.35 800	. . . . 5 ⊢ ((∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
18	2, 16, 17	sylancl 585	. . . 4 ⊢ (𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)
19	18	ax-gen 1789	. . 3 ⊢ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)
20		dfss2 3960	. . . 4 ⊢ (∪ {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴))
21	20	biimpri 227	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴) → ∪ {𝐴} ⊆ 𝐴)
22	19, 21	ax-mp 5	. 2 ⊢ ∪ {𝐴} ⊆ 𝐴
23		id 22	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
24		unisnALT.1	. . . . . 6 ⊢ 𝐴 ∈ V
25	24	snid 4656	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
26		elunii 4904	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ ∪ {𝐴})
27	23, 25, 26	sylancl 585	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})
28	27	ax-gen 1789	. . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})
29		dfss2 3960	. . . 4 ⊢ (𝐴 ⊆ ∪ {𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}))
30	29	biimpri 227	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) → 𝐴 ⊆ ∪ {𝐴})
31	28, 30	ax-mp 5	. 2 ⊢ 𝐴 ⊆ ∪ {𝐴}
32	22, 31	eqssi 3990	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3940  {csn 4620  ∪ cuni 4899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-ss 3957  df-sn 4621  df-uni 4900

This theorem is referenced by: (None)