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Metamath Proof Explorer

Theorem engch

Description: The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015)

Ref Expression
Assertion engch ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ GCH ↔ 𝐵 ∈ GCH ) )

Proof

Step Hyp Ref Expression
1 enfi ⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ Fin ↔ 𝐵 ∈ Fin ) )
2 sdomen1 ⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ≺ 𝑥 ↔ 𝐵 ≺ 𝑥 ) )
3 pwen ⊢ ( 𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵 )
4 sdomen2 ⊢ ( 𝒫 𝐴 ≈ 𝒫 𝐵 → ( 𝑥 ≺ 𝒫 𝐴 ↔ 𝑥 ≺ 𝒫 𝐵 ) )
5 3 4 syl ⊢ ( 𝐴 ≈ 𝐵 → ( 𝑥 ≺ 𝒫 𝐴 ↔ 𝑥 ≺ 𝒫 𝐵 ) )
6 2 5 anbi12d ⊢ ( 𝐴 ≈ 𝐵 → ( ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ↔ ( 𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵 ) ) )
7 6 notbid ⊢ ( 𝐴 ≈ 𝐵 → ( ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ↔ ¬ ( 𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵 ) ) )
8 7 albidv ⊢ ( 𝐴 ≈ 𝐵 → ( ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ↔ ∀ 𝑥 ¬ ( 𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵 ) ) )
9 1 8 orbi12d ⊢ ( 𝐴 ≈ 𝐵 → ( ( 𝐴 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ) ↔ ( 𝐵 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵 ) ) ) )
10 relen ⊢ Rel ≈
11 10 brrelex1i ⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ∈ V )
12 elgch ⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ GCH ↔ ( 𝐴 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ) ) )
13 11 12 syl ⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ GCH ↔ ( 𝐴 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ) ) )
14 10 brrelex2i ⊢ ( 𝐴 ≈ 𝐵 → 𝐵 ∈ V )
15 elgch ⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ GCH ↔ ( 𝐵 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵 ) ) ) )
16 14 15 syl ⊢ ( 𝐴 ≈ 𝐵 → ( 𝐵 ∈ GCH ↔ ( 𝐵 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵 ) ) ) )
17 9 13 16 3bitr4d ⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ GCH ↔ 𝐵 ∈ GCH ) )