Theorem n0abso 44966

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 44959	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤)))
2		tru 1544	. . . 4 ⊢ ⊤
3	2	rext0 44937	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ 𝐴 ≠ ∅)
4	3	bicomi 224	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ⊤)
5	2	biantru 529	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	5	rexbii 3093	. 2 ⊢ (∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤))
7	1, 4, 6	3bitr4g 314	1 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ⊤wtru 1541   ∈ wcel 2108   ≠ wne 2939  ∃wrex 3069  ∅c0 4332  Tr wtr 5257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-ss 3967  df-nul 4333  df-uni 4906  df-tr 5258

This theorem is referenced by:  modelac8prim  44982