Theorem unisnALT 41632

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 41632 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 41632. 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 41632, 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 4803	. . . . . 6 ⊢ (𝑥 ∈ ∪ {𝐴} ↔ ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}))
2	1	biimpi 219	. . . . 5 ⊢ (𝑥 ∈ ∪ {𝐴} → ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}))
3		id 22	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → (𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}))
4		simpl 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞)
6		simpr 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
7	3, 6	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴})
8		elsni 4542	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝐴} → 𝑞 = 𝐴)
9	7, 8	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 = 𝐴)
10		eleq2 2878	. . . . . . . . 9 ⊢ (𝑞 = 𝐴 → (𝑥 ∈ 𝑞 ↔ 𝑥 ∈ 𝐴))
11	10	biimpac 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 = 𝐴) → 𝑥 ∈ 𝐴)
12	5, 9, 11	syl2anc 587	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
13	12	ax-gen 1797	. . . . . 6 ⊢ ∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)
14		19.23v 1943	. . . . . . 7 ⊢ (∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) ↔ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴))
15	14	biimpi 219	. . . . . 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		pm3.35 802	. . . . 5 ⊢ ((∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
18	2, 16, 17	sylancl 589	. . . 4 ⊢ (𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)
19	18	ax-gen 1797	. . 3 ⊢ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)
20		dfss2 3901	. . . 4 ⊢ (∪ {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴))
21	20	biimpri 231	. . 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 4561	. . . . 5 ⊢ 𝐴 ∈ {𝐴}
26		elunii 4805	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ ∪ {𝐴})
27	23, 25, 26	sylancl 589	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})
28	27	ax-gen 1797	. . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})
29		dfss2 3901	. . . 4 ⊢ (𝐴 ⊆ ∪ {𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}))
30	29	biimpri 231	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) → 𝐴 ⊆ ∪ {𝐴})
31	28, 30	ax-mp 5	. 2 ⊢ 𝐴 ⊆ ∪ {𝐴}
32	22, 31	eqssi 3931	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  {csn 4525  ∪ cuni 4800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-sn 4526  df-uni 4801

This theorem is referenced by: (None)