Theorem iotanul 5302

Description: Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotanul	⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Proof of Theorem iotanul

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2082	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		dfiota2 5287	. . . 4 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
3		alnex 1547	. . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
4		ax-in2 620	. . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
5	4	alimi 1503	. . . . . . . . 9 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6		ss2ab 3295	. . . . . . . . 9 ⊢ ({𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧} ↔ ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
7	5, 6	sylibr 134	. . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧})
8		dfnul2 3496	. . . . . . . 8 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
9	7, 8	sseqtrrdi 3276	. . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
10	3, 9	sylbir 135	. . . . . 6 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
11	10	unissd 3917	. . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∪ ∅)
12		uni0 3920	. . . . 5 ⊢ ∪ ∅ = ∅
13	11, 12	sseqtrdi 3275	. . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
14	2, 13	eqsstrid 3273	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
15	1, 14	sylnbi 684	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16		ss0 3535	. 2 ⊢ ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
17	15, 16	syl 14	1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1395   = wceq 1397  ∃wex 1540  ∃!weu 2079  {cab 2217   ⊆ wss 3200  ∅c0 3494  ∪ 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-in1 619  ax-in2 620  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-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-uni 3894  df-iota 5286

This theorem is referenced by:  tz6.12-2  5630  0fv  5677  riotaund  6008  0g0  13461