Theorem el 5405

Description: Any set is an element of some other set. See elALT 5409 for a shorter proof using more axioms, and see elALT2 5326 for a proof that uses ax-9 2152 and ax-pow 5322 instead of ax-pr 5390. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5390 instead of ax-9 2152 and ax-pow 5322. (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 5390	. 2 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦)
2		orc 878	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑥 ∨ 𝑧 = 𝑥))
3		ax8v1 2146	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦))
4	2, 3	embantd 59	. . 3 ⊢ (𝑧 = 𝑥 → (((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦))
5	4	spimvw 2006	. 2 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
6	1, 5	eximii 1857	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 858  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-8 2144  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1800

This theorem is referenced by:  sels  5407  dmep  5899  elirrvOLD  9546  axpownd  10559  zfcndinf  10576  distel  36151  axtco1from2  36835