Metamath Proof Explorer


Theorem prfi

Description: An unordered pair is finite. For a shorter proof using ax-un , see prfiALT . (Contributed by NM, 22-Aug-2008) Avoid ax-11 , ax-un . (Revised by BTernaryTau, 13-Jan-2025)

Ref Expression
Assertion prfi A B Fin

Proof

Step Hyp Ref Expression
1 prprc1 ¬ A V A B = B
2 snfi B Fin
3 1 2 eqeltrdi ¬ A V A B Fin
4 prprc2 ¬ B V A B = A
5 snfi A Fin
6 4 5 eqeltrdi ¬ B V A B Fin
7 2onn 2 𝑜 ω
8 simp1 A V B V ¬ A = B A V
9 simp2 A V B V ¬ A = B B V
10 simp3 A V B V ¬ A = B ¬ A = B
11 8 9 10 enpr2d A V B V ¬ A = B A B 2 𝑜
12 breq2 x = 2 𝑜 A B x A B 2 𝑜
13 12 rspcev 2 𝑜 ω A B 2 𝑜 x ω A B x
14 7 11 13 sylancr A V B V ¬ A = B x ω A B x
15 isfi A B Fin x ω A B x
16 14 15 sylibr A V B V ¬ A = B A B Fin
17 16 3expia A V B V ¬ A = B A B Fin
18 dfsn2 A = A A
19 preq2 A = B A A = A B
20 18 19 eqtr2id A = B A B = A
21 20 5 eqeltrdi A = B A B Fin
22 17 21 pm2.61d2 A V B V A B Fin
23 3 6 22 ecase A B Fin