Theorem eliota 5278

Description: An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)

Assertion

Ref	Expression
eliota	⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Distinct variable groups:   𝜑,𝑦   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem eliota

Step	Hyp	Ref	Expression
1		dfiota2 5252	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2	1	eleq2i 2274	. 2 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ 𝐴 ∈ ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)})
3		eluniab 3876	. 2 ⊢ (𝐴 ∈ ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
4	2, 3	bitri 184	1 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1371  ∃wex 1516   ∈ wcel 2178  {cab 2193  ∪ cuni 3864  ℩cio 5249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sn 3649  df-uni 3865  df-iota 5251

This theorem is referenced by:  eliotaeu  5279