Theorem iotassuniOLD 6523

Description: Obsolete version of iotassuni 6516 as of 23-Dec-2024. (Contributed by Mario Carneiro, 24-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
iotassuniOLD	⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Proof of Theorem iotassuniOLD

Step	Hyp	Ref	Expression
1		iotauni 6519	. . 3 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		eqimss 4041	. . 3 ⊢ ((℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑} → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	syl 17	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
4		iotanul 6522	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5		0ss 4397	. . 3 ⊢ ∅ ⊆ ∪ {𝑥 ∣ 𝜑}
6	4, 5	eqsstrdi 4037	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
7	3, 6	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1540  ∃!weu 2561  {cab 2708   ⊆ wss 3949  ∅c0 4323  ∪ cuni 4909  ℩cio 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496

This theorem is referenced by: (None)