Theorem unisnALT 42546

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 42546 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 42546. 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 42546, 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 4842	. . . . . 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 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞)
6		simpr 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
7	3, 6	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8		elsni 4578	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
9	7, 8	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10		eleq2 2827	. . . . . . . . 9 ⊢ (𝑞 = 𝐴 → (𝑥 ∈ 𝑞 ↔ 𝑥 ∈ 𝐴))
11	10	biimpac 479	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 = 𝐴) → 𝑥 ∈ 𝐴)
12	5, 9, 11	syl2anc 584	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
13	12	ax-gen 1798	. . . . . 6 ⊢ ∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
14		19.23v 1945	. . . . . . 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 586	. . . 4 ⊢ (𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)
19	18	ax-gen 1798	. . 3 ⊢ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)
20		dfss2 3907	. . . 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 4597	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
26		elunii 4844	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ ∪ {𝐴})
27	23, 25, 26	sylancl 586	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})
28	27	ax-gen 1798	. . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})
29		dfss2 3907	. . . 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 3937	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  {csn 4561  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-sn 4562  df-uni 4840

This theorem is referenced by: (None)