Theorem n0abso 45718

Description: Nonemptiness is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)

Assertion

Ref	Expression
n0abso	⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴))

Distinct variable groups:   𝑥,𝑀   𝑥,𝐴

Proof of Theorem n0abso

Step	Hyp	Ref	Expression
1		rexabso 45711	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤)))
2		tru 1574	. . . 4 ⊢ ⊤
3	2	rext0 45680	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ 𝐴 ≠ ∅)
4	3	bicomi 227	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ⊤)
5	2	biantru 539	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	5	rexbii 3114	. 2 ⊢ (∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤))
7	1, 4, 6	3bitr4g 317	1 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ⊤wtru 1571   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  ∅c0 4286  Tr wtr 5220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-v 3459  df-dif 3909  df-ss 3923  df-nul 4287  df-uni 4875  df-tr 5221

This theorem is used by:  modelac8prim  45734