Theorem rex2dom 6995

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 2814	. . 3 |- ( A e. V -> A e. _V )
2		prssi 3831	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> { x , y } C_ A )
3		df2o3 6596	. . . . . . . 8 |- 2o = { (/) , 1o }
4		0ex 4216	. . . . . . . . . 10 |- (/) e. _V
5	4	a1i 9	. . . . . . . . 9 |- ( x =/= y -> (/) e. _V )
6		1oex 6589	. . . . . . . . . 10 |- 1o e. _V
7	6	a1i 9	. . . . . . . . 9 |- ( x =/= y -> 1o e. _V )
8		vex 2805	. . . . . . . . . 10 |- x e. _V
9	8	a1i 9	. . . . . . . . 9 |- ( x =/= y -> x e. _V )
10		vex 2805	. . . . . . . . . 10 |- y e. _V
11	10	a1i 9	. . . . . . . . 9 |- ( x =/= y -> y e. _V )
12		1n0 6599	. . . . . . . . . . 11 |- 1o =/= (/)
13	12	necomi 2487	. . . . . . . . . 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 6991	. . . . . . . 8 |- ( x =/= y -> { (/) , 1o } ~~ { x , y } )
17	3, 16	eqbrtrid 4123	. . . . . . 7 |- ( x =/= y -> 2o ~~ { x , y } )
18		endom 6935	. . . . . . 7 |- ( 2o ~~ { x , y } -> 2o ~<_ { x , y } )
19	17, 18	syl 14	. . . . . 6 |- ( x =/= y -> 2o ~<_ { x , y } )
20		domssr 6950	. . . . . . 7 |- ( ( A e. _V /\ { x , y } C_ A /\ 2o ~<_ { x , y } ) -> 2o ~<_ A )
21	20	3expib 1232	. . . . . 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 2657	. . 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 2202   =/= wne 2402   E.wrex 2511   _Vcvv 2802   C_ wss 3200   (/)c0 3494   {cpr 3670   class class class wbr 4088   1oc1o 6574   2oc2o 6575   ~~ cen 6906   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-1o 6581  df-2o 6582  df-en 6909  df-dom 6910

This theorem is referenced by:  hashdmprop2dom  11107  fun2dmnop0  11110