Theorem n0abso 46073

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 46066	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤)))
2		tru 1573	. . . 4 ⊢ ⊤
3	2	rext0 46028	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ 𝐴 ≠ ∅)
4	3	bicomi 227	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ⊤)
5	2	biantru 539	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	5	rexbii 3109	. 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 1570   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  ∅c0 4277  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3897  df-dif 4046  df-nul 4278  df-uni 4867  df-tr 5212

This theorem is used by:  modelac8prim  46089