Theorem 2dom 4427

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 4141	. . . 4 |- 2o = {(/), {(/)}}
2	1	breq1i 2626	. . 3 |- (2o ~<_ A <-> {(/), {(/)}} ~<_ A)
3		2dom.1	. . . 4 |- A e. V
4	3	brdom 4378	. . 3 |- ({(/), {(/)}} ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
5	2, 4	bitr 173	. 2 |- (2o ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
6		eqeq1 1481	. . . . . 6 |- (x = (f` (/)) -> (x = y <-> (f` (/)) = y))
7	6	negbid 611	. . . . 5 |- (x = (f` (/)) -> (-. x = y <-> -. (f` (/)) = y))
8		eqeq2 1484	. . . . . 6 |- (y = (f` {(/)}) -> ((f` (/)) = y <-> (f` (/)) = (f` {(/)})))
9	8	negbid 611	. . . . 5 |- (y = (f` {(/)}) -> (-. (f` (/)) = y <-> -. (f` (/)) = (f` {(/)})))
10	7, 9	rcla42ev 1881	. . . 4 |- (((f` (/)) e. A /\ (f` {(/)}) e. A /\ -. (f` (/)) = (f` {(/)})) -> E.x e. A E.y e. A -. x = y)
11		f1f 3665	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> f:{(/), {(/)}}-->A)
12		0ex 2711	. . . . . . 7 |- (/) e. V
13	12	pri1 2450	. . . . . 6 |- (/) e. {(/), {(/)}}
14		ffvelrn 3814	. . . . . 6 |- ((f:{(/), {(/)}}-->A /\ (/) e. {(/), {(/)}}) -> (f` (/)) e. A)
15	13, 14	mpan2 696	. . . . 5 |- (f:{(/), {(/)}}-->A -> (f` (/)) e. A)
16	11, 15	syl 10	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` (/)) e. A)
17		p0ex 2770	. . . . . . 7 |- {(/)} e. V
18	17	pri2 2451	. . . . . 6 |- {(/)} e. {(/), {(/)}}
19		ffvelrn 3814	. . . . . 6 |- ((f:{(/), {(/)}}-->A /\ {(/)} e. {(/), {(/)}}) -> (f` {(/)}) e. A)
20	18, 19	mpan2 696	. . . . 5 |- (f:{(/), {(/)}}-->A -> (f` {(/)}) e. A)
21	11, 20	syl 10	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` {(/)}) e. A)
22		0nep0 2737	. . . . . 6 |- (/) =/= {(/)}
23		df-ne 1587	. . . . . 6 |- ((/) =/= {(/)} <-> -. (/) = {(/)})
24	22, 23	mpbi 189	. . . . 5 |- -. (/) = {(/)}
25	13, 18	pm3.2i 285	. . . . . 6 |- ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})
26		f1fveq 3876	. . . . . 6 |- ((f:{(/), {(/)}}-1-1->A /\ ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})) -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
27	25, 26	mpan2 696	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
28	24, 27	mtbiri 717	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> -. (f` (/)) = (f` {(/)}))
29	10, 16, 21, 28	syl3anc 858	. . 3 |- (f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
30	29	19.23aiv 1295	. 2 |- (E.f f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
31	5, 30	sylbi 199	1 |- (2o ~<_ A -> E.x e. A E.y e. A -. x = y)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  E.wrex 1646  Vcvv 1811  (/)c0 2280  {csn 2409  {cpr 2410   class class class wbr 2619  -->wf 3178  -1-1->wf1 3179  ` cfv 3182  2oc2o 4129   ~<_ cdom 4365

This theorem is referenced by:  unxpdomlem 4843

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fv 3198  df-1o 4133  df-2o 4134  df-dom 4369

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