Theorem iotanul 5270

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 2060	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		dfiota2 5255	. . . 4 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
3		alnex 1525	. . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
4		ax-in2 618	. . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
5	4	alimi 1481	. . . . . . . . 9 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6		ss2ab 3272	. . . . . . . . 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 3473	. . . . . . . 8 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
9	7, 8	sseqtrrdi 3253	. . . . . . 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 3891	. . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∪ ∅)
12		uni0 3894	. . . . 5 ⊢ ∪ ∅ = ∅
13	11, 12	sseqtrdi 3252	. . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} ⊆ ∅)
14	2, 13	eqsstrid 3250	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
15	1, 14	sylnbi 682	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16		ss0 3512	. 2 ⊢ ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
17	15, 16	syl 14	1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1373   = wceq 1375  ∃wex 1518  ∃!weu 2057  {cab 2195   ⊆ wss 3177  ∅c0 3471  ∪ cuni 3867  ℩cio 5252

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-dif 3179  df-in 3183  df-ss 3190  df-nul 3472  df-sn 3652  df-uni 3868  df-iota 5254

This theorem is referenced by:  tz6.12-2  5594  0fv  5639  riotaund  5964  0g0  13375