Theorem el 5400

Description: Any set is an element of some other set. See elALT 5403 for a shorter proof using more axioms, and see elALT2 5327 for a proof that uses ax-9 2119 and ax-pow 5323 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 2119 and ax-pow 5323. (Revised by BTernaryTau, 2-Dec-2024.)

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		pm4.25 905	. . . . . 6 ⊢ (𝑧 = 𝑥 ↔ (𝑧 = 𝑥 ∨ 𝑧 = 𝑥))
3	2	imbi1i 349	. . . . 5 ⊢ ((𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦))
4	3	albii 1819	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦))
5		elequ1 2116	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
6	5	equsalvw 2004	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
7	4, 6	bitr3i 277	. . 3 ⊢ (∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
8	7	exbii 1848	. 2 ⊢ (∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 = 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦 𝑥 ∈ 𝑦)
9	1, 8	mpbi 230	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:   → wi 4   ∨ wo 847  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780

This theorem is referenced by:  sels  5401  dtruOLD  5404  dmep  5890  axpownd  10561  zfcndinf  10578  distel  35798