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Theorem iotanul 4825
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 1900 . . 3 (∃!xφzx(φx = z))
2 dfiota2 4811 . . . 4 (℩xφ) = {zx(φx = z)}
3 alnex 1385 . . . . . . 7 (z ¬ x(φx = z) ↔ ¬ zx(φx = z))
4 ax-in2 545 . . . . . . . . . 10 x(φx = z) → (x(φx = z) → ¬ z = z))
54alimi 1341 . . . . . . . . 9 (z ¬ x(φx = z) → z(x(φx = z) → ¬ z = z))
6 ss2ab 3002 . . . . . . . . 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 3220 . . . . . . . 8 ∅ = {z ∣ ¬ z = z}
97, 8syl6sseqr 2986 . . . . . . 7 (z ¬ x(φx = z) → {zx(φx = z)} ⊆ ∅)
103, 9sylbir 125 . . . . . 6 zx(φx = z) → {zx(φx = z)} ⊆ ∅)
1110unissd 3595 . . . . 5 zx(φx = z) → {zx(φx = z)} ⊆ ∅)
12 uni0 3598 . . . . 5 ∅ = ∅
1311, 12syl6sseq 2985 . . . 4 zx(φx = z) → {zx(φx = z)} ⊆ ∅)
142, 13syl5eqss 2983 . . 3 zx(φx = z) → (℩xφ) ⊆ ∅)
151, 14sylnbi 602 . 2 ∃!xφ → (℩xφ) ⊆ ∅)
16 ss0 3251 . 2 ((℩xφ) ⊆ ∅ → (℩xφ) = ∅)
1715, 16syl 14 1 ∃!xφ → (℩xφ) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1240   = wceq 1242  wex 1378  ∃!weu 1897  {cab 2023  wss 2911  c0 3218   cuni 3571  cio 4808
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572  df-iota 4810
This theorem is referenced by:  tz6.12-2  5112  0fv  5151  riotaund  5445
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