Theorem 2dom 4414

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 4131	. . . 4 |- 2o = {(/), {(/)}}
2	1	breq1i 2621	. . 3 |- (2o ~<_ A <-> {(/), {(/)}} ~<_ A)
3		2dom.1	. . . 4 |- A e. V
4	3	brdom 4366	. . 3 |- ({(/), {(/)}} ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
5	2, 4	bitr 173	. 2 |- (2o ~<_ A <-> E.f f:{(/), {(/)}}-1-1->A)
6		eqeq1 1478	. . . . . 6 |- (x = (f` (/)) -> (x = y <-> (f` (/)) = y))
7	6	negbid 610	. . . . 5 |- (x = (f` (/)) -> (-. x = y <-> -. (f` (/)) = y))
8		eqeq2 1481	. . . . . 6 |- (y = (f` {(/)}) -> ((f` (/)) = y <-> (f` (/)) = (f` {(/)})))
9	8	negbid 610	. . . . 5 |- (y = (f` {(/)}) -> (-. (f` (/)) = y <-> -. (f` (/)) = (f` {(/)})))
10	7, 9	rcla42ev 1877	. . . 4 |- (((f` (/)) e. A /\ (f` {(/)}) e. A /\ -. (f` (/)) = (f` {(/)})) -> E.x e. A E.y e. A -. x = y)
11		f1f 3656	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> f:{(/), {(/)}}-->A)
12		0ex 2706	. . . . . . 7 |- (/) e. V
13	12	pri1 2446	. . . . . 6 |- (/) e. {(/), {(/)}}
14		ffvelrn 3805	. . . . . 6 |- ((f:{(/), {(/)}}-->A /\ (/) e. {(/), {(/)}}) -> (f` (/)) e. A)
15	13, 14	mpan2 695	. . . . 5 |- (f:{(/), {(/)}}-->A -> (f` (/)) e. A)
16	11, 15	syl 10	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` (/)) e. A)
17		p0ex 2765	. . . . . . 7 |- {(/)} e. V
18	17	pri2 2447	. . . . . 6 |- {(/)} e. {(/), {(/)}}
19		ffvelrn 3805	. . . . . 6 |- ((f:{(/), {(/)}}-->A /\ {(/)} e. {(/), {(/)}}) -> (f` {(/)}) e. A)
20	18, 19	mpan2 695	. . . . 5 |- (f:{(/), {(/)}}-->A -> (f` {(/)}) e. A)
21	11, 20	syl 10	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> (f` {(/)}) e. A)
22		0nep0 2732	. . . . . 6 |- (/) =/= {(/)}
23		df-ne 1584	. . . . . 6 |- ((/) =/= {(/)} <-> -. (/) = {(/)})
24	22, 23	mpbi 189	. . . . 5 |- -. (/) = {(/)}
25	13, 18	pm3.2i 285	. . . . . 6 |- ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})
26		f1fveq 3867	. . . . . 6 |- ((f:{(/), {(/)}}-1-1->A /\ ((/) e. {(/), {(/)}} /\ {(/)} e. {(/), {(/)}})) -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
27	25, 26	mpan2 695	. . . . 5 |- (f:{(/), {(/)}}-1-1->A -> ((f` (/)) = (f` {(/)}) <-> (/) = {(/)}))
28	24, 27	mtbiri 716	. . . 4 |- (f:{(/), {(/)}}-1-1->A -> -. (f` (/)) = (f` {(/)}))
29	10, 16, 21, 28	syl3anc 857	. . 3 |- (f:{(/), {(/)}}-1-1->A -> E.x e. A E.y e. A -. x = y)
30	29	19.23aiv 1293	. 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 954   e. wcel 956  E.wex 978   =/= wne 1582  E.wrex 1643  Vcvv 1807  (/)c0 2276  {csn 2405  {cpr 2406   class class class wbr 2614  -->wf 3173  -1-1->wf1 3174  ` cfv 3177  2oc2o 4119   ~<_ cdom 4355

This theorem is referenced by:  unxpdomlem 4823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fv 3193  df-1o 4123  df-2o 4124  df-dom 4358

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