Theorem php2

Description: Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998) Avoid ax-pow . (Revised by BTernaryTau, 20-Nov-2024)

	Ref	Expression
Assertion	php2	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → 𝐵 ≺ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nnfi	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin )
2		pssss	⊢ ( 𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴 )
3		ssdomfi	⊢ ( 𝐴 ∈ Fin → ( 𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴 ) )
4	3	imp	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ≼ 𝐴 )
5	1 2 4	syl2an	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → 𝐵 ≼ 𝐴 )
6		php	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 )
7		ensymfib	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴 ) )
8	7	biimprd	⊢ ( 𝐴 ∈ Fin → ( 𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵 ) )
9	1 8	syl	⊢ ( 𝐴 ∈ ω → ( 𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵 ) )
10	9	adantr	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ( 𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵 ) )
11	6 10	mtod	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ¬ 𝐵 ≈ 𝐴 )
12		brsdom	⊢ ( 𝐵 ≺ 𝐴 ↔ ( 𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴 ) )
13	5 11 12	sylanbrc	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → 𝐵 ≺ 𝐴 )