Theorem iotaexOLD 6522

Description: Obsolete version of iotaex 6515 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
iotaexOLD	⊢ (℩𝑥𝜑) ∈ V

Proof of Theorem iotaexOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 6513	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
2	1	eqcomd 2740	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
3	2	eximi 1834	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4		eu6 2572	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
5		isset 3478	. . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 292	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7		iotanul 6520	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8		0ex 5289	. . 3 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2841	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
10	6, 9	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃!weu 2566  Vcvv 3464  ∅c0 4315  ℩cio 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-nul 5288

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-v 3466  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-sn 4609  df-pr 4611  df-uni 4890  df-iota 6495

This theorem is referenced by: (None)