Metamath Proof Explorer


Theorem unidifsnel

Description: The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017)

Ref Expression
Assertion unidifsnel XPP2𝑜PXP

Proof

Step Hyp Ref Expression
1 2onn 2𝑜ω
2 nnfi 2𝑜ω2𝑜Fin
3 1 2 ax-mp 2𝑜Fin
4 enfi P2𝑜PFin2𝑜Fin
5 3 4 mpbiri P2𝑜PFin
6 5 adantl XPP2𝑜PFin
7 diffi PFinPXFin
8 6 7 syl XPP2𝑜PXFin
9 8 cardidd XPP2𝑜cardPXPX
10 9 ensymd XPP2𝑜PXcardPX
11 simpl XPP2𝑜XP
12 dif1card PFinXPcardP=succardPX
13 6 11 12 syl2anc XPP2𝑜cardP=succardPX
14 cardennn P2𝑜2𝑜ωcardP=2𝑜
15 1 14 mpan2 P2𝑜cardP=2𝑜
16 df-2o 2𝑜=suc1𝑜
17 15 16 eqtrdi P2𝑜cardP=suc1𝑜
18 17 adantl XPP2𝑜cardP=suc1𝑜
19 13 18 eqtr3d XPP2𝑜succardPX=suc1𝑜
20 suc11reg succardPX=suc1𝑜cardPX=1𝑜
21 19 20 sylib XPP2𝑜cardPX=1𝑜
22 10 21 breqtrd XPP2𝑜PX1𝑜
23 en1 PX1𝑜xPX=x
24 22 23 sylib XPP2𝑜xPX=x
25 simpr XPP2𝑜PX=xPX=x
26 25 unieqd XPP2𝑜PX=xPX=x
27 unisnv x=x
28 26 27 eqtrdi XPP2𝑜PX=xPX=x
29 difssd XPP2𝑜PX=xPXP
30 25 29 eqsstrrd XPP2𝑜PX=xxP
31 vsnid xx
32 ssel2 xPxxxP
33 30 31 32 sylancl XPP2𝑜PX=xxP
34 28 33 eqeltrd XPP2𝑜PX=xPXP
35 24 34 exlimddv XPP2𝑜PXP