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Theorem iotanul 5071
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.
StepHypRef Expression
1 df-eu 1978 . . 3 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 5057 . . . 4 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1458 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 ax-in2 587 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
54alimi 1414 . . . . . . . . 9 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6 ss2ab 3133 . . . . . . . . 9 ({𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧} ↔ ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
75, 6sylibr 133 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧})
8 dfnul2 3333 . . . . . . . 8 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
97, 8sseqtrrdi 3114 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
103, 9sylbir 134 . . . . . 6 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
1110unissd 3728 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
12 uni0 3731 . . . . 5 ∅ = ∅
1311, 12sseqtrdi 3113 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
142, 13eqsstrid 3111 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
151, 14sylnbi 650 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16 ss0 3371 . 2 ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
1715, 16syl 14 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wal 1312   = wceq 1314  wex 1451  ∃!weu 1975  {cab 2101  wss 3039  c0 3331   cuni 3704  cio 5054
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-uni 3705  df-iota 5056
This theorem is referenced by:  tz6.12-2  5378  0fv  5422  riotaund  5730
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