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Mirrors > Home > MPE Home > Th. List > Mathboxes > unisnALT | Structured version Visualization version GIF version |
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 41280 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 41280. 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 41280, 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.) |
Ref | Expression |
---|---|
unisnALT.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
unisnALT | ⊢ ∪ {𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 4841 | . . . . . 6 ⊢ (𝑥 ∈ ∪ {𝐴} ↔ ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴})) | |
2 | 1 | biimpi 218 | . . . . 5 ⊢ (𝑥 ∈ ∪ {𝐴} → ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴})) |
3 | id 22 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → (𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴})) | |
4 | simpl 485 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞) |
6 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴}) | |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴}) |
8 | elsni 4584 | . . . . . . . . 9 ⊢ (𝑞 ∈ {𝐴} → 𝑞 = 𝐴) | |
9 | 7, 8 | syl 17 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 = 𝐴) |
10 | eleq2 2901 | . . . . . . . . 9 ⊢ (𝑞 = 𝐴 → (𝑥 ∈ 𝑞 ↔ 𝑥 ∈ 𝐴)) | |
11 | 10 | biimpac 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 = 𝐴) → 𝑥 ∈ 𝐴) |
12 | 5, 9, 11 | syl2anc 586 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) |
13 | 12 | ax-gen 1796 | . . . . . 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 218 | . . . . . 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 | pm3.35 801 | . . . . 5 ⊢ ((∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴) | |
18 | 2, 16, 17 | sylancl 588 | . . . 4 ⊢ (𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴) |
19 | 18 | ax-gen 1796 | . . 3 ⊢ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴) |
20 | dfss2 3955 | . . . 4 ⊢ (∪ {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)) | |
21 | 20 | biimpri 230 | . . 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 4601 | . . . . 5 ⊢ 𝐴 ∈ {𝐴} |
26 | elunii 4843 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ ∪ {𝐴}) | |
27 | 23, 25, 26 | sylancl 588 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) |
28 | 27 | ax-gen 1796 | . . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) |
29 | dfss2 3955 | . . . 4 ⊢ (𝐴 ⊆ ∪ {𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})) | |
30 | 29 | biimpri 230 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) → 𝐴 ⊆ ∪ {𝐴}) |
31 | 28, 30 | ax-mp 5 | . 2 ⊢ 𝐴 ⊆ ∪ {𝐴} |
32 | 22, 31 | eqssi 3983 | 1 ⊢ ∪ {𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {csn 4567 ∪ cuni 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-sn 4568 df-uni 4839 |
This theorem is referenced by: (None) |
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