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Metamath Proof Explorer

Theorem inf3lemb

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 for detailed description. (Contributed by NM, 28-Oct-1996)

Ref Expression
Hypotheses inf3lem.1 ⊢ G = y ∈ V ⟼ w ∈ x | w ∩ x ⊆ y
inf3lem.2 ⊢ F = rec ⁡ G ∅ ↾ ω
inf3lem.3 ⊢ A ∈ V
inf3lem.4 ⊢ B ∈ V
Assertion inf3lemb ⊢ F ⁡ ∅ = ∅

Proof

Step Hyp Ref Expression
1 inf3lem.1 ⊢ G = y ∈ V ⟼ w ∈ x | w ∩ x ⊆ y
2 inf3lem.2 ⊢ F = rec ⁡ G ∅ ↾ ω
3 inf3lem.3 ⊢ A ∈ V
4 inf3lem.4 ⊢ B ∈ V
5 2 fveq1i ⊢ F ⁡ ∅ = rec ⁡ G ∅ ↾ ω ⁡ ∅
6 0ex ⊢ ∅ ∈ V
7 fr0g ⊢ ∅ ∈ V → rec ⁡ G ∅ ↾ ω ⁡ ∅ = ∅
8 6 7 ax-mp ⊢ rec ⁡ G ∅ ↾ ω ⁡ ∅ = ∅
9 5 8 eqtri ⊢ F ⁡ ∅ = ∅