Theorem el 5421

Description: Any set is an element of some other set. See elALT 5425 for a shorter proof using more axioms, and see elALT2 5342 for a proof that uses ax-9 2156 and ax-pow 5338 instead of ax-pr 5406. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5406 instead of ax-9 2156 and ax-pow 5338. (Revised by BTernaryTau, 2-Dec-2024.) (Proof shortened by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
el	⊢ ∃𝑦 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem el

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pr 5406	. 2 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)
2		orc 881	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑥 ∨ 𝑧 = 𝑥))
3		ax8v1 2150	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦))
4	2, 3	embantd 60	. . 3 ⊢ (𝑧 = 𝑥 → (((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦))
5	4	spimvw 2019	. 2 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
6	1, 5	eximii 1870	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-8 2148  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813

This theorem is used by:  sels  5423  dmep  5915  elirrvOLD  9567  axpownd  10601  zfcndinf  10618  distel  36330  axtco1from2  37043