Theorem el 5420

Description: Any set is an element of some other set. See elALT 5424 for a shorter proof using more axioms, and see elALT2 5341 for a proof that uses ax-9 2159 and ax-pow 5337 instead of ax-pr 5405. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5405 instead of ax-9 2159 and ax-pow 5337. (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 5405	. 2 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)
2		orc 880	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑥 ∨ 𝑧 = 𝑥))
3		ax8v1 2153	. . . 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 2013	. 2 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
6	1, 5	eximii 1864	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 860  ∀wal 1565  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-8 2151  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1807

This theorem is referenced by:  sels  5422  dmep  5914  elirrvOLD  9559  axpownd  10585  zfcndinf  10602  distel  36191  axtco1from2  36874