Theorem 2dom 4488

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 4199	. . . 4 |- 2o = {(/), {(/)}}
2	1	breq1i 2681	. . 3 |- (2o ~<_ A <-> {(/), {(/)}} ~<_ A)
3		2dom.1	. . . 4 |- A e. V
4	3	brdom 4439	. . 3 |- ({(/), {(/)}} ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
5	2, 4	bitri 180	. 2 |- (2o ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
6		eqeq1 1528	. . . . . 6 |- (x = (f` (/)) -> (x = y <-> (f` (/)) = y))
7	6	notbid 622	. . . . 5 |- (x = (f` (/)) -> (-. x = y <-> -. (f` (/)) = y))
8		eqeq2 1531	. . . . . 6 |- (y = (f` {(/)}) -> ((f` (/)) = y <-> (f` (/)) = (f` {(/)})))
9	8	notbid 622	. . . . 5 |- (y = (f` {(/)}) -> (-. (f` (/)) = y <-> -. (f` (/)) = (f` {(/)})))
10	7, 9	rcla42ev 1928	. . . 4 |- (((f` (/)) e. A /\ (f` {(/)}) e. A /\ -. (f` (/)) = (f` {(/)})) -> E.x e. A E.y e. A -. x = y)
11		f1f 3722	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> f:{(/), {(/)}}-->A)
12		0ex 2766	. . . . . . 7 |- (/) e. V
13	12	prid1 2504	. . . . . 6 |- (/) e. {(/), {(/)}}
14		ffvelrn 3871	. . . . . 6 |- ((f:{(/), {(/)}}-->A /\ (/) e. {(/), {(/)}}) -> (f` (/)) e. A)
15	13, 14	mpan2 708	. . . . 5 |- (f:{(/), {(/)}}-->A -> (f` (/)) e. A)
16	11, 15	syl 10	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` (/)) e. A)
17		p0ex 2826	. . . . . . 7 |- {(/)} e. V
18	17	prid2 2505	. . . . . 6 |- {(/)} e. {(/), {(/)}}
19		ffvelrn 3871	. . . . . 6 |- ((f:{(/), {(/)}}-->A /\ {(/)} e. {(/), {(/)}}) -> (f` {(/)}) e. A)
20	18, 19	mpan2 708	. . . . 5 |- (f:{(/), {(/)}}-->A -> (f` {(/)}) e. A)
21	11, 20	syl 10	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` {(/)}) e. A)
22		0nep0 2792	. . . . . 6 |- (/) =/= {(/)}
23		df-ne 1634	. . . . . 6 |- ((/) =/= {(/)} <-> -. (/) = {(/)})
24	22, 23	mpbi 196	. . . . 5 |- -. (/) = {(/)}
25	13, 18	pm3.2i 292	. . . . . 6 |- ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})
26		f1fveq 3934	. . . . . 6 |- ((f:{(/), {(/)}}-1-1->A /\ ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})) -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
27	25, 26	mpan2 708	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
28	24, 27	mtbiri 729	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> -. (f` (/)) = (f` {(/)}))
29	10, 16, 21, 28	syl3anc 870	. . 3 |- (f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
30	29	19.23aiv 1337	. 2 |- (E.f f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
31	5, 30	sylbi 206	1 |- (2o ~<_ A -> E.x e. A E.y e. A -. x = y)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999  E.wex 1021   =/= wne 1632  E.wrex 1693  Vcvv 1858  (/)c0 2331  {csn 2461  {cpr 2462   class class class wbr 2674  -->wf 3235  -1-1->wf1 3236  ` cfv 3239  2oc2o 4187   ~<_ cdom 4426

This theorem is referenced by:  unxpdomlem 4908

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-suc 3011  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-f1 3252  df-fv 3255  df-1o 4191  df-2o 4192  df-dom 4430

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