Theorem iotaex 4357

Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotaex	⊢ (℩xφ) ∈ V

Proof of Theorem iotaex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 4351	. . . . 5 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z)
2	1	eqcomd 2358	. . . 4 ⊢ (∀x(φ ↔ x = z) → z = (℩xφ))
3	2	eximi 1576	. . 3 ⊢ (∃z∀x(φ ↔ x = z) → ∃z z = (℩xφ))
4		df-eu 2208	. . 3 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
5		isset 2864	. . 3 ⊢ ((℩xφ) ∈ V ↔ ∃z z = (℩xφ))
6	3, 4, 5	3imtr4i 257	. 2 ⊢ (∃!xφ → (℩xφ) ∈ V)
7		iotanul 4355	. . 3 ⊢ (¬ ∃!xφ → (℩xφ) = ∅)
8		0ex 4111	. . 3 ⊢ ∅ ∈ V
9	7, 8	syl6eqel 2441	. 2 ⊢ (¬ ∃!xφ → (℩xφ) ∈ V)
10	6, 9	pm2.61i 156	1 ⊢ (℩xφ) ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860  ∅c0 3551  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  iota4an  4359  ncfinex  4473  tfinex  4486  fvex  5340  tcex  6158