Theorem dvdemo1 5239

Description: Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑦 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑧).

That theorem bundles the theorems (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑥 ∈ 𝑥) with 𝑥, 𝑦 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥) with 𝑥, 𝑦 disjoint).

Compare with dvdemo2 5240, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5240 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.)

Note that dvdemo1 5239 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑧 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require ax-11 2158 nor ax-13 2379. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

Assertion

Ref	Expression
dvdemo1	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem dvdemo1

Step	Hyp	Ref	Expression
1		dtru 5236	. . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		exnal 1828	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbir 234	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
4		pm2.21 123	. 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
5	3, 4	eximii 1838	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781

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-12 2175  ax-nul 5174  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)