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Theorem dfiota4 4372
 Description: Alternate definition of iota in terms of 1c. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
dfiota4 (℩xφ) = (1c ∩ {{x φ}})

Proof of Theorem dfiota4
StepHypRef Expression
1 iotauni 4351 . . 3 (∃!xφ → (℩xφ) = {x φ})
2 dfeu2 4333 . . . . . . . 8 (∃!xφ ↔ {x φ} 1c)
3 snssi 3852 . . . . . . . 8 ({x φ} 1c → {{x φ}} 1c)
42, 3sylbi 187 . . . . . . 7 (∃!xφ → {{x φ}} 1c)
5 df-ss 3259 . . . . . . . 8 ({{x φ}} 1c ↔ ({{x φ}} ∩ 1c) = {{x φ}})
6 incom 3448 . . . . . . . . 9 ({{x φ}} ∩ 1c) = (1c ∩ {{x φ}})
76eqeq1i 2360 . . . . . . . 8 (({{x φ}} ∩ 1c) = {{x φ}} ↔ (1c ∩ {{x φ}}) = {{x φ}})
85, 7bitri 240 . . . . . . 7 ({{x φ}} 1c ↔ (1c ∩ {{x φ}}) = {{x φ}})
94, 8sylib 188 . . . . . 6 (∃!xφ → (1c ∩ {{x φ}}) = {{x φ}})
109unieqd 3902 . . . . 5 (∃!xφ(1c ∩ {{x φ}}) = {{x φ}})
11 euabex 4334 . . . . . 6 (∃!xφ → {x φ} V)
12 unisng 3908 . . . . . 6 ({x φ} V → {{x φ}} = {x φ})
1311, 12syl 15 . . . . 5 (∃!xφ{{x φ}} = {x φ})
1410, 13eqtrd 2385 . . . 4 (∃!xφ(1c ∩ {{x φ}}) = {x φ})
1514unieqd 3902 . . 3 (∃!xφ(1c ∩ {{x φ}}) = {x φ})
161, 15eqtr4d 2388 . 2 (∃!xφ → (℩xφ) = (1c ∩ {{x φ}}))
17 iotanul 4354 . . 3 ∃!xφ → (℩xφ) = )
182notbii 287 . . . . . . . 8 ∃!xφ ↔ ¬ {x φ} 1c)
19 disjsn 3786 . . . . . . . 8 ((1c ∩ {{x φ}}) = ↔ ¬ {x φ} 1c)
2018, 19bitr4i 243 . . . . . . 7 ∃!xφ ↔ (1c ∩ {{x φ}}) = )
2120biimpi 186 . . . . . 6 ∃!xφ → (1c ∩ {{x φ}}) = )
2221unieqd 3902 . . . . 5 ∃!xφ(1c ∩ {{x φ}}) = )
2322unieqd 3902 . . . 4 ∃!xφ(1c ∩ {{x φ}}) = )
24 uni0 3918 . . . . . 6 =
2524unieqi 3901 . . . . 5 =
2625, 24eqtri 2373 . . . 4 =
2723, 26syl6eq 2401 . . 3 ∃!xφ(1c ∩ {{x φ}}) = )
2817, 27eqtr4d 2388 . 2 ∃!xφ → (℩xφ) = (1c ∩ {{x φ}}))
2916, 28pm2.61i 156 1 (℩xφ) = (1c ∩ {{x φ}})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  {csn 3737  ∪cuni 3891  1cc1c 4134  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-nin 4078  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-1c 4136  df-iota 4339 This theorem is referenced by: (None)
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