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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 39162 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 39162. 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 39162, 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 4439 | . . . . . 6 ⊢ (𝑥 ∈ ∪ {𝐴} ↔ ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴})) | |
2 | 1 | biimpi 206 | . . . . 5 ⊢ (𝑥 ∈ ∪ {𝐴} → ∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴})) |
3 | id 22 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → (𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴})) | |
4 | simpl 473 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝑞) |
6 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴}) | |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 ∈ {𝐴}) |
8 | elsni 4194 | . . . . . . . . 9 ⊢ (𝑞 ∈ {𝐴} → 𝑞 = 𝐴) | |
9 | 7, 8 | syl 17 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑞 = 𝐴) |
10 | eleq2 2690 | . . . . . . . . 9 ⊢ (𝑞 = 𝐴 → (𝑥 ∈ 𝑞 ↔ 𝑥 ∈ 𝐴)) | |
11 | 10 | biimpac 503 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 = 𝐴) → 𝑥 ∈ 𝐴) |
12 | 5, 9, 11 | syl2anc 693 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) |
13 | 12 | ax-gen 1722 | . . . . . 6 ⊢ ∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) |
14 | 19.23v 1902 | . . . . . . 7 ⊢ (∀𝑞((𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴) ↔ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)) | |
15 | 14 | biimpi 206 | . . . . . 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 | pm3.35 611 | . . . . 5 ⊢ ((∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) ∧ (∃𝑞(𝑥 ∈ 𝑞 ∧ 𝑞 ∈ {𝐴}) → 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴) | |
18 | 2, 16, 17 | sylancl 694 | . . . 4 ⊢ (𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴) |
19 | 18 | ax-gen 1722 | . . 3 ⊢ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴) |
20 | dfss2 3591 | . . . 4 ⊢ (∪ {𝐴} ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ {𝐴} → 𝑥 ∈ 𝐴)) | |
21 | 20 | biimpri 218 | . . 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 4208 | . . . . 5 ⊢ 𝐴 ∈ {𝐴} |
26 | elunii 4441 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ ∪ {𝐴}) | |
27 | 23, 25, 26 | sylancl 694 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) |
28 | 27 | ax-gen 1722 | . . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) |
29 | dfss2 3591 | . . . 4 ⊢ (𝐴 ⊆ ∪ {𝐴} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴})) | |
30 | 29 | biimpri 218 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝐴}) → 𝐴 ⊆ ∪ {𝐴}) |
31 | 28, 30 | ax-mp 5 | . 2 ⊢ 𝐴 ⊆ ∪ {𝐴} |
32 | 22, 31 | eqssi 3619 | 1 ⊢ ∪ {𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {csn 4177 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-sn 4178 df-uni 4437 |
This theorem is referenced by: (None) |
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