Theorem rex2dom 7100

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 2833	. . 3 |- ( A e. V -> A e. _V )
2		prssi 3868	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> { x , y } C_ A )
3		df2o3 6692	. . . . . . . 8 |- 2o = { (/) , 1o }
4		0ex 4255	. . . . . . . . . 10 |- (/) e. _V
5	4	a1i 9	. . . . . . . . 9 |- ( x =/= y -> (/) e. _V )
6		1oex 6685	. . . . . . . . . 10 |- 1o e. _V
7	6	a1i 9	. . . . . . . . 9 |- ( x =/= y -> 1o e. _V )
8		vex 2824	. . . . . . . . . 10 |- x e. _V
9	8	a1i 9	. . . . . . . . 9 |- ( x =/= y -> x e. _V )
10		vex 2824	. . . . . . . . . 10 |- y e. _V
11	10	a1i 9	. . . . . . . . 9 |- ( x =/= y -> y e. _V )
12		1n0 6695	. . . . . . . . . . 11 |- 1o =/= (/)
13	12	necomi 2505	. . . . . . . . . 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 7096	. . . . . . . 8 |- ( x =/= y -> { (/) , 1o } ~~ { x , y } )
17	3, 16	eqbrtrid 4160	. . . . . . 7 |- ( x =/= y -> 2o ~~ { x , y } )
18		endom 7039	. . . . . . 7 |- ( 2o ~~ { x , y } -> 2o ~<_ { x , y } )
19	17, 18	syl 14	. . . . . 6 |- ( x =/= y -> 2o ~<_ { x , y } )
20		domssr 7054	. . . . . . 7 |- ( ( A e. _V /\ { x , y } C_ A /\ 2o ~<_ { x , y } ) -> 2o ~<_ A )
21	20	3expib 1237	. . . . . 6 |- ( A e. _V -> ( ( { x , y } C_ A /\ 2o ~<_ { x , y } ) -> 2o ~<_ A ) )
22	2, 19, 21	syl2ani 412	. . . . 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 2675	. . 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 2209   =/= wne 2420   E.wrex 2529   _Vcvv 2821   C_ wss 3220   (/)c0 3520   {cpr 3706   class class class wbr 4125   1oc1o 6670   2oc2o 6671   ~~ cen 7010   ~<_ cdom 7011

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-1o 6677  df-2o 6678  df-en 7013  df-dom 7014

This theorem is referenced by:  hashdmprop2dom  11274  fun2dmnop0  11280