Theorem iotanul 5111

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 2003	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		dfiota2 5097	. . . 4 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
3		alnex 1476	. . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
4		ax-in2 605	. . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
5	4	alimi 1432	. . . . . . . . 9 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6		ss2ab 3170	. . . . . . . . 9 ⊢ ({𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧} ↔ ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
7	5, 6	sylibr 133	. . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧})
8		dfnul2 3370	. . . . . . . 8 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
9	7, 8	sseqtrrdi 3151	. . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
10	3, 9	sylbir 134	. . . . . 6 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
11	10	unissd 3768	. . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∪ ∅)
12		uni0 3771	. . . . 5 ⊢ ∪ ∅ = ∅
13	11, 12	sseqtrdi 3150	. . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
14	2, 13	eqsstrid 3148	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
15	1, 14	sylnbi 668	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16		ss0 3408	. 2 ⊢ ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
17	15, 16	syl 14	1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469  ∃!weu 2000  {cab 2126   ⊆ wss 3076  ∅c0 3368  ∪ cuni 3744  ℩cio 5094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-uni 3745  df-iota 5096

This theorem is referenced by:  tz6.12-2  5420  0fv  5464  riotaund  5772