Theorem fisshasheq

Description: A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023)

	Ref	Expression
Assertion	fisshasheq	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssfi	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ∈ Fin )
2	1	3adant3	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → 𝐴 ∈ Fin )
3		hashen	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )
4	3	biimp3a	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → 𝐴 ≈ 𝐵 )
5		pm3.2	⊢ ( 𝐵 ∈ Fin → ( 𝐴 ⊆ 𝐵 → ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ) ) )
6	5	3ad2ant2	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → ( 𝐴 ⊆ 𝐵 → ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ) ) )
7		fisseneq	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵 ) → 𝐴 = 𝐵 )
8	7	3expa	⊢ ( ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ) ∧ 𝐴 ≈ 𝐵 ) → 𝐴 = 𝐵 )
9	8	expcom	⊢ ( 𝐴 ≈ 𝐵 → ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 = 𝐵 ) )
10	4 6 9	sylsyld	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → ( 𝐴 ⊆ 𝐵 → 𝐴 = 𝐵 ) )
11	10	3expb	⊢ ( ( 𝐴 ∈ Fin ∧ ( 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) ) → ( 𝐴 ⊆ 𝐵 → 𝐴 = 𝐵 ) )
12	11	expcom	⊢ ( ( 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → ( 𝐴 ∈ Fin → ( 𝐴 ⊆ 𝐵 → 𝐴 = 𝐵 ) ) )
13	12	com23	⊢ ( ( 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∈ Fin → 𝐴 = 𝐵 ) ) )
14	13	3impia	⊢ ( ( 𝐵 ∈ Fin ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∈ Fin → 𝐴 = 𝐵 ) )
15	14	3com23	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → ( 𝐴 ∈ Fin → 𝐴 = 𝐵 ) )
16	2 15	mpd	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ) → 𝐴 = 𝐵 )