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Theorem iotanul 4809
 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 1885 . . 3 (∃!xφzx(φx = z))
2 dfiota2 4795 . . . 4 (℩xφ) = {zx(φx = z)}
3 alnex 1369 . . . . . . 7 (z ¬ x(φx = z) ↔ ¬ zx(φx = z))
4 ax-in2 533 . . . . . . . . . 10 x(φx = z) → (x(φx = z) → ¬ z = z))
54alimi 1324 . . . . . . . . 9 (z ¬ x(φx = z) → z(x(φx = z) → ¬ z = z))
6 ss2ab 2985 . . . . . . . . 9 ({zx(φx = z)} ⊆ {z ∣ ¬ z = z} ↔ z(x(φx = z) → ¬ z = z))
75, 6sylibr 137 . . . . . . . 8 (z ¬ x(φx = z) → {zx(φx = z)} ⊆ {z ∣ ¬ z = z})
8 dfnul2 3203 . . . . . . . 8 ∅ = {z ∣ ¬ z = z}
97, 8syl6sseqr 2969 . . . . . . 7 (z ¬ x(φx = z) → {zx(φx = z)} ⊆ ∅)
103, 9sylbir 125 . . . . . 6 zx(φx = z) → {zx(φx = z)} ⊆ ∅)
1110unissd 3578 . . . . 5 zx(φx = z) → {zx(φx = z)} ⊆ ∅)
12 uni0 3581 . . . . 5 ∅ = ∅
1311, 12syl6sseq 2968 . . . 4 zx(φx = z) → {zx(φx = z)} ⊆ ∅)
142, 13syl5eqss 2966 . . 3 zx(φx = z) → (℩xφ) ⊆ ∅)
151, 14sylnbi 590 . 2 ∃!xφ → (℩xφ) ⊆ ∅)
16 ss0 3234 . 2 ((℩xφ) ⊆ ∅ → (℩xφ) = ∅)
1715, 16syl 14 1 ∃!xφ → (℩xφ) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1226   = wceq 1228  ∃wex 1362  ∃!weu 1882  {cab 2008   ⊆ wss 2894  ∅c0 3201  ∪ cuni 3554  ℩cio 4792 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-uni 3555  df-iota 4794 This theorem is referenced by:  tz6.12-2  5094  0fv  5133  riotaund  5426
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