Theorem n0abso 45513

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 45506	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤)))
2		tru 1563	. . . 4 ⊢ ⊤
3	2	rext0 45475	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ 𝐴 ≠ ∅)
4	3	bicomi 226	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ⊤)
5	2	biantru 537	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	5	rexbii 3108	. 2 ⊢ (∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ ⊤))
7	1, 4, 6	3bitr4g 316	1 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ⊤wtru 1560   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  ∅c0 4283  Tr wtr 5204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-v 3455  df-dif 3905  df-ss 3919  df-nul 4284  df-uni 4863  df-tr 5205

This theorem is referenced by:  modelac8prim  45529