Theorem ralf0 3656

Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Hypothesis

Ref	Expression
ralf0.1	⊢ ¬ φ

Assertion

Ref	Expression
ralf0	⊢ (∀x ∈ A φ ↔ A = ∅)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem ralf0

Step	Hyp	Ref	Expression
1		ralf0.1	. . . . 5 ⊢ ¬ φ
2		con3 126	. . . . 5 ⊢ ((x ∈ A → φ) → (¬ φ → ¬ x ∈ A))
3	1, 2	mpi 16	. . . 4 ⊢ ((x ∈ A → φ) → ¬ x ∈ A)
4	3	alimi 1559	. . 3 ⊢ (∀x(x ∈ A → φ) → ∀x ¬ x ∈ A)
5		df-ral 2619	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		eq0 3564	. . 3 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)
7	4, 5, 6	3imtr4i 257	. 2 ⊢ (∀x ∈ A φ → A = ∅)
8		rzal 3651	. 2 ⊢ (A = ∅ → ∀x ∈ A φ)
9	7, 8	impbii 180	1 ⊢ (∀x ∈ A φ ↔ A = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∅c0 3550

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

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by: (None)