Theorem iotanul 4355

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 2208	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
2		dfiota2 4341	. . 3 ⊢ (℩xφ) = ∪{z ∣ ∀x(φ ↔ x = z)}
3		alnex 1543	. . . . . 6 ⊢ (∀z ¬ ∀x(φ ↔ x = z) ↔ ¬ ∃z∀x(φ ↔ x = z))
4		ax-1 6	. . . . . . . . . . 11 ⊢ (¬ ∀x(φ ↔ x = z) → (z = z → ¬ ∀x(φ ↔ x = z)))
5		eqidd 2354	. . . . . . . . . . 11 ⊢ (¬ ∀x(φ ↔ x = z) → z = z)
6	4, 5	impbid1 194	. . . . . . . . . 10 ⊢ (¬ ∀x(φ ↔ x = z) → (z = z ↔ ¬ ∀x(φ ↔ x = z)))
7	6	con2bid 319	. . . . . . . . 9 ⊢ (¬ ∀x(φ ↔ x = z) → (∀x(φ ↔ x = z) ↔ ¬ z = z))
8	7	alimi 1559	. . . . . . . 8 ⊢ (∀z ¬ ∀x(φ ↔ x = z) → ∀z(∀x(φ ↔ x = z) ↔ ¬ z = z))
9		abbi 2464	. . . . . . . 8 ⊢ (∀z(∀x(φ ↔ x = z) ↔ ¬ z = z) ↔ {z ∣ ∀x(φ ↔ x = z)} = {z ∣ ¬ z = z})
10	8, 9	sylib 188	. . . . . . 7 ⊢ (∀z ¬ ∀x(φ ↔ x = z) → {z ∣ ∀x(φ ↔ x = z)} = {z ∣ ¬ z = z})
11		dfnul2 3553	. . . . . . 7 ⊢ ∅ = {z ∣ ¬ z = z}
12	10, 11	syl6eqr 2403	. . . . . 6 ⊢ (∀z ¬ ∀x(φ ↔ x = z) → {z ∣ ∀x(φ ↔ x = z)} = ∅)
13	3, 12	sylbir 204	. . . . 5 ⊢ (¬ ∃z∀x(φ ↔ x = z) → {z ∣ ∀x(φ ↔ x = z)} = ∅)
14	13	unieqd 3903	. . . 4 ⊢ (¬ ∃z∀x(φ ↔ x = z) → ∪{z ∣ ∀x(φ ↔ x = z)} = ∪∅)
15		uni0 3919	. . . 4 ⊢ ∪∅ = ∅
16	14, 15	syl6eq 2401	. . 3 ⊢ (¬ ∃z∀x(φ ↔ x = z) → ∪{z ∣ ∀x(φ ↔ x = z)} = ∅)
17	2, 16	syl5eq 2397	. 2 ⊢ (¬ ∃z∀x(φ ↔ x = z) → (℩xφ) = ∅)
18	1, 17	sylnbi 297	1 ⊢ (¬ ∃!xφ → (℩xφ) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  ∅c0 3551  ∪cuni 3892  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  iotassuni  4356  iotaex  4357  dfiota3  4371  dfiota4  4373  fvprc  5326