Theorem eliota 5306

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 5279	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2	1	eleq2i 2296	. 2 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ 𝐴 ∈ ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)})
3		eluniab 3900	. 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 1393  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∪ cuni 3888  ℩cio 5276

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sn 3672  df-uni 3889  df-iota 5278

This theorem is referenced by:  eliotaeu  5307