Theorem iotaexOLD 6523

Description: Obsolete version of iotaex 6516 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 6514	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
2	1	eqcomd 2731	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
3	2	eximi 1829	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4		eu6 2562	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
5		isset 3476	. . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 291	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7		iotanul 6521	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8		0ex 5302	. . 3 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2833	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
10	6, 9	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃!weu 2556  Vcvv 3463  ∅c0 4318  ℩cio 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-nul 5301

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-sn 4625  df-pr 4627  df-uni 4904  df-iota 6495

This theorem is referenced by: (None)