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Metamath Proof Explorer

Theorem fiinfg

Description: Lemma showing existence and closure of infimum of a finite set. (Contributed by AV, 6-Oct-2020)

Ref Expression
Assertion fiinfg ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 fiming ⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) )
2 equcom ⊢ ( 𝑥 = 𝑦 ↔ 𝑦 = 𝑥 )
3 sotrieq2 ⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) → ( 𝑦 = 𝑥 ↔ ( ¬ 𝑦 𝑅 𝑥 ∧ ¬ 𝑥 𝑅 𝑦 ) ) )
4 3 ancom2s ⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑦 = 𝑥 ↔ ( ¬ 𝑦 𝑅 𝑥 ∧ ¬ 𝑥 𝑅 𝑦 ) ) )
5 2 4 bitrid ⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 = 𝑦 ↔ ( ¬ 𝑦 𝑅 𝑥 ∧ ¬ 𝑥 𝑅 𝑦 ) ) )
6 5 simprbda ⊢ ( ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) ∧ 𝑥 = 𝑦 ) → ¬ 𝑦 𝑅 𝑥 )
7 6 ex ⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 = 𝑦 → ¬ 𝑦 𝑅 𝑥 ) )
8 7 anassrs ⊢ ( ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 = 𝑦 → ¬ 𝑦 𝑅 𝑥 ) )
9 8 a1dd ⊢ ( ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 = 𝑦 → ( ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ¬ 𝑦 𝑅 𝑥 ) ) )
10 pm2.27 ⊢ ( 𝑥 ≠ 𝑦 → ( ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → 𝑥 𝑅 𝑦 ) )
11 soasym ⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝑅 𝑦 → ¬ 𝑦 𝑅 𝑥 ) )
12 11 anassrs ⊢ ( ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 → ¬ 𝑦 𝑅 𝑥 ) )
13 10 12 syl9r ⊢ ( ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ≠ 𝑦 → ( ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ¬ 𝑦 𝑅 𝑥 ) ) )
14 9 13 pm2.61dne ⊢ ( ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ¬ 𝑦 𝑅 𝑥 ) )
15 14 ralimdva ⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 𝑅 𝑥 ) )
16 breq1 ⊢ ( 𝑧 = 𝑥 → ( 𝑧 𝑅 𝑦 ↔ 𝑥 𝑅 𝑦 ) )
17 16 rspcev ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 )
18 17 ex ⊢ ( 𝑥 ∈ 𝐴 → ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) )
19 18 ralrimivw ⊢ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) )
20 19 adantl ⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) )
21 15 20 jctird ⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) ) ) )
22 21 reximdva ⊢ ( 𝑅 Or 𝐴 → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) ) ) )
23 22 3ad2ant1 ⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑥 𝑅 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) ) ) )
24 1 23 mpd ⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 𝑅 𝑦 ) ) )