Theorem eliota 5314

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 5287	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2	1	eleq2i 2298	. 2 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ 𝐴 ∈ ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)})
3		eluniab 3905	. 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 1395  ∃wex 1540   ∈ wcel 2202  {cab 2217  ∪ cuni 3893  ℩cio 5284

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sn 3675  df-uni 3894  df-iota 5286

This theorem is referenced by:  eliotaeu  5315