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Metamath Proof Explorer

Theorem extssr

Description: Property of subset relation, see also extid , extep and the comment of df-ssr . (Contributed by Peter Mazsa, 10-Jul-2019)

Ref Expression
Assertion extssr ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( [ 𝐴 ] ◡ S = [ 𝐵 ] ◡ S ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 brssr ⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 S 𝐴 ↔ 𝑥 ⊆ 𝐴 ) )
2 brssr ⊢ ( 𝐵 ∈ 𝑊 → ( 𝑥 S 𝐵 ↔ 𝑥 ⊆ 𝐵 ) )
3 1 2 bi2bian9 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝑥 S 𝐴 ↔ 𝑥 S 𝐵 ) ↔ ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) ) )
4 3 albidv ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ( 𝑥 S 𝐴 ↔ 𝑥 S 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) ) )
5 relssr ⊢ Rel S
6 releccnveq ⊢ ( ( Rel S ∧ Rel S ) → ( [ 𝐴 ] ◡ S = [ 𝐵 ] ◡ S ↔ ∀ 𝑥 ( 𝑥 S 𝐴 ↔ 𝑥 S 𝐵 ) ) )
7 5 5 6 mp2an ⊢ ( [ 𝐴 ] ◡ S = [ 𝐵 ] ◡ S ↔ ∀ 𝑥 ( 𝑥 S 𝐴 ↔ 𝑥 S 𝐵 ) )
8 ssext ⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) )
9 4 7 8 3bitr4g ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( [ 𝐴 ] ◡ S = [ 𝐵 ] ◡ S ↔ 𝐴 = 𝐵 ) )