Theorem 2dom 5486

Description: A set that dominates ordinal 2 has at least 2 different members.

Hypothesis

Ref	Expression
2dom.1	|- A e. _V

Assertion

Ref	Expression
2dom	|- (2o ~<_ A -> E.x e. A E.y e. A -. x = y)

Distinct variable group:   x,y,A

Proof of Theorem 2dom

Step	Hyp	Ref	Expression
1		df2o2 5186	. . . 4 |- 2o = {(/), {(/)}}
2	1	breq1i 3345	. . 3 |- (2o ~<_ A <-> {(/), {(/)}} ~<_ A)
3		2dom.1	. . . 4 |- A e. _V
4	3	brdom 5437	. . 3 |- ({(/), {(/)}} ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
5	2, 4	bitri 190	. 2 |- (2o ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
6		ffvelrn 4787	. . . . 5 |- ((f:{(/), {(/)}}-->A /\ (/) e. {(/), {(/)}}) -> (f` (/)) e. A)
7		f1f 4610	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> f:{(/), {(/)}}-->A)
8		0ex 3446	. . . . . 6 |- (/) e. _V
9	8	prid1 3106	. . . . 5 |- (/) e. {(/), {(/)}}
10	6, 7, 9	sylancl 525	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` (/)) e. A)
11		ffvelrn 4787	. . . . 5 |- ((f:{(/), {(/)}}-->A /\ {(/)} e. {(/), {(/)}}) -> (f` {(/)}) e. A)
12		p0ex 3495	. . . . . 6 |- {(/)} e. _V
13	12	prid2 3107	. . . . 5 |- {(/)} e. {(/), {(/)}}
14	11, 7, 13	sylancl 525	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` {(/)}) e. A)
15		0nep0 3474	. . . . . 6 |- (/) =/= {(/)}
16		df-ne 2019	. . . . . 6 |- ((/) =/= {(/)} <-> -. (/) = {(/)})
17	15, 16	mpbi 206	. . . . 5 |- -. (/) = {(/)}
18		f1fveq 4852	. . . . . 6 |- ((f:{(/), {(/)}}-1-1->A /\ ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})) -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
19	9, 13, 18	mpanr12 778	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
20	17, 19	mtbiri 785	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> -. (f` (/)) = (f` {(/)}))
21		eqeq1 1890	. . . . . 6 |- (x = (f` (/)) -> (x = y <-> (f` (/)) = y))
22	21	notbid 673	. . . . 5 |- (x = (f` (/)) -> (-. x = y <-> -. (f` (/)) = y))
23		eqeq2 1893	. . . . . 6 |- (y = (f` {(/)}) -> ((f` (/)) = y <-> (f` (/)) = (f` {(/)})))
24	23	notbid 673	. . . . 5 |- (y = (f` {(/)}) -> (-. (f` (/)) = y <-> -. (f` (/)) = (f` {(/)})))
25	22, 24	rcla42ev 2385	. . . 4 |- (((f` (/)) e. A /\ (f` {(/)}) e. A /\ -. (f` (/)) = (f` {(/)})) -> E.x e. A E.y e. A -. x = y)
26	10, 14, 20, 25	syl111anc 1100	. . 3 |- (f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
27	26	19.23aiv 1674	. 2 |- (E.f f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
28	5, 27	sylbi 216	1 |- (2o ~<_ A -> E.x e. A E.y e. A -. x = y)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  E.wrex 2106  _Vcvv 2292  (/)c0 2875  {csn 3044  {cpr 3045   class class class wbr 3338  -->wf 3994  -1-1->wf1 3995  ` cfv 3998  2oc2o 5173   ~<_ cdom 5424

This theorem is referenced by:  unxpdomlem 5995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fv 4014  df-1o 5177  df-2o 5178  df-dom 5428

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