Theorem rex2dom 7076

Description: A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)

Assertion

Ref	Expression
rex2dom	|- ( ( A e. V /\ E. x e. A E. y e. A x =/= y ) -> 2o ~<_ A )

Distinct variable group:   x, A, y

Allowed substitution hints:   V( x, y)

Proof of Theorem rex2dom

Step	Hyp	Ref	Expression
1		elex 2827	. . 3 |- ( A e. V -> A e. _V )
2		prssi 3857	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> { x , y } C_ A )
3		df2o3 6675	. . . . . . . 8 |- 2o = { (/) , 1o }
4		0ex 4242	. . . . . . . . . 10 |- (/) e. _V
5	4	a1i 9	. . . . . . . . 9 |- ( x =/= y -> (/) e. _V )
6		1oex 6668	. . . . . . . . . 10 |- 1o e. _V
7	6	a1i 9	. . . . . . . . 9 |- ( x =/= y -> 1o e. _V )
8		vex 2818	. . . . . . . . . 10 |- x e. _V
9	8	a1i 9	. . . . . . . . 9 |- ( x =/= y -> x e. _V )
10		vex 2818	. . . . . . . . . 10 |- y e. _V
11	10	a1i 9	. . . . . . . . 9 |- ( x =/= y -> y e. _V )
12		1n0 6678	. . . . . . . . . . 11 |- 1o =/= (/)
13	12	necomi 2499	. . . . . . . . . 10 |- (/) =/= 1o
14	13	a1i 9	. . . . . . . . 9 |- ( x =/= y -> (/) =/= 1o )
15		id 19	. . . . . . . . 9 |- ( x =/= y -> x =/= y )
16	5, 7, 9, 11, 14, 15	en2prd 7072	. . . . . . . 8 |- ( x =/= y -> { (/) , 1o } ~~ { x , y } )
17	3, 16	eqbrtrid 4149	. . . . . . 7 |- ( x =/= y -> 2o ~~ { x , y } )
18		endom 7015	. . . . . . 7 |- ( 2o ~~ { x , y } -> 2o ~<_ { x , y } )
19	17, 18	syl 14	. . . . . 6 |- ( x =/= y -> 2o ~<_ { x , y } )
20		domssr 7030	. . . . . . 7 |- ( ( A e. _V /\ { x , y } C_ A /\ 2o ~<_ { x , y } ) -> 2o ~<_ A )
21	20	3expib 1233	. . . . . 6 |- ( A e. _V -> ( ( { x , y } C_ A /\ 2o ~<_ { x , y } ) -> 2o ~<_ A ) )
22	2, 19, 21	syl2ani 408	. . . . 5 |- ( A e. _V -> ( ( ( x e. A /\ y e. A ) /\ x =/= y ) -> 2o ~<_ A ) )
23	22	expd 258	. . . 4 |- ( A e. _V -> ( ( x e. A /\ y e. A ) -> ( x =/= y -> 2o ~<_ A ) ) )
24	23	rexlimdvv 2669	. . 3 |- ( A e. _V -> ( E. x e. A E. y e. A x =/= y -> 2o ~<_ A ) )
25	1, 24	syl 14	. 2 |- ( A e. V -> ( E. x e. A E. y e. A x =/= y -> 2o ~<_ A ) )
26	25	imp 124	1 |- ( ( A e. V /\ E. x e. A E. y e. A x =/= y ) -> 2o ~<_ A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   =/= wne 2414   E.wrex 2523   _Vcvv 2815   C_ wss 3214   (/)c0 3512   {cpr 3695   class class class wbr 4114   1oc1o 6653   2oc2o 6654   ~~ cen 6986   ~<_ cdom 6987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-1o 6660  df-2o 6661  df-en 6989  df-dom 6990

This theorem is referenced by:  hashdmprop2dom  11241  fun2dmnop0  11247