Theorem 3dom 16901

Description: A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.)

Assertion

Ref	Expression
3dom	|- ( 3o ~<_ A -> E. x e. A E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) )

Distinct variable group:   x, y, z, A

Proof of Theorem 3dom

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6999	. 2 |- ( 3o ~<_ A -> E. f f : 3o -1-1-> A )
2		f1f 5578	. . . . 5 |- ( f : 3o -1-1-> A -> f : 3o --> A )
3	2	adantl 277	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> f : 3o --> A )
4		0lt2o 6687	. . . . . . 7 |- (/) e. 2o
5		elelsuc 4535	. . . . . . 7 |- ( (/) e. 2o -> (/) e. suc 2o )
6	4, 5	ax-mp 5	. . . . . 6 |- (/) e. suc 2o
7		df-3o 6662	. . . . . 6 |- 3o = suc 2o
8	6, 7	eleqtrri 2310	. . . . 5 |- (/) e. 3o
9	8	a1i 9	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> (/) e. 3o )
10	3, 9	ffvelcdmd 5818	. . 3 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` (/) ) e. A )
11		1lt2o 6688	. . . . . . . 8 |- 1o e. 2o
12		elelsuc 4535	. . . . . . . 8 |- ( 1o e. 2o -> 1o e. suc 2o )
13	11, 12	ax-mp 5	. . . . . . 7 |- 1o e. suc 2o
14	13, 7	eleqtrri 2310	. . . . . 6 |- 1o e. 3o
15	14	a1i 9	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> 1o e. 3o )
16	3, 15	ffvelcdmd 5818	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 1o ) e. A )
17		2onn 6767	. . . . . . . . . 10 |- 2o e. om
18	17	elexi 2828	. . . . . . . . 9 |- 2o e. _V
19	18	sucid 4543	. . . . . . . 8 |- 2o e. suc 2o
20	19, 7	eleqtrri 2310	. . . . . . 7 |- 2o e. 3o
21	20	a1i 9	. . . . . 6 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> 2o e. 3o )
22	3, 21	ffvelcdmd 5818	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 2o ) e. A )
23		1n0 6678	. . . . . . . . . 10 |- 1o =/= (/)
24	23	nesymi 2460	. . . . . . . . 9 |- -. (/) = 1o
25		f1veqaeq 5948	. . . . . . . . 9 |- ( ( f : 3o -1-1-> A /\ ( (/) e. 3o /\ 1o e. 3o ) ) -> ( ( f ` (/) ) = ( f ` 1o ) -> (/) = 1o ) )
26	24, 25	mtoi 670	. . . . . . . 8 |- ( ( f : 3o -1-1-> A /\ ( (/) e. 3o /\ 1o e. 3o ) ) -> -. ( f ` (/) ) = ( f ` 1o ) )
27	8, 14, 26	mpanr12 439	. . . . . . 7 |- ( f : 3o -1-1-> A -> -. ( f ` (/) ) = ( f ` 1o ) )
28	27	neqned 2421	. . . . . 6 |- ( f : 3o -1-1-> A -> ( f ` (/) ) =/= ( f ` 1o ) )
29	28	adantl 277	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` (/) ) =/= ( f ` 1o ) )
30		2on0 6670	. . . . . . . . . 10 |- 2o =/= (/)
31	30	nesymi 2460	. . . . . . . . 9 |- -. (/) = 2o
32		f1veqaeq 5948	. . . . . . . . 9 |- ( ( f : 3o -1-1-> A /\ ( (/) e. 3o /\ 2o e. 3o ) ) -> ( ( f ` (/) ) = ( f ` 2o ) -> (/) = 2o ) )
33	31, 32	mtoi 670	. . . . . . . 8 |- ( ( f : 3o -1-1-> A /\ ( (/) e. 3o /\ 2o e. 3o ) ) -> -. ( f ` (/) ) = ( f ` 2o ) )
34	8, 20, 33	mpanr12 439	. . . . . . 7 |- ( f : 3o -1-1-> A -> -. ( f ` (/) ) = ( f ` 2o ) )
35	34	neqned 2421	. . . . . 6 |- ( f : 3o -1-1-> A -> ( f ` (/) ) =/= ( f ` 2o ) )
36	35	adantl 277	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` (/) ) =/= ( f ` 2o ) )
37		nnord 4739	. . . . . . . . . . . . 13 |- ( 2o e. om -> Ord 2o )
38	17, 37	ax-mp 5	. . . . . . . . . . . 12 |- Ord 2o
39		ordirr 4669	. . . . . . . . . . . 12 |- ( Ord 2o -> -. 2o e. 2o )
40	38, 39	ax-mp 5	. . . . . . . . . . 11 |- -. 2o e. 2o
41		eleq1 2297	. . . . . . . . . . 11 |- ( 1o = 2o -> ( 1o e. 2o <-> 2o e. 2o ) )
42	40, 41	mtbiri 682	. . . . . . . . . 10 |- ( 1o = 2o -> -. 1o e. 2o )
43	11, 42	mt2 645	. . . . . . . . 9 |- -. 1o = 2o
44		f1veqaeq 5948	. . . . . . . . 9 |- ( ( f : 3o -1-1-> A /\ ( 1o e. 3o /\ 2o e. 3o ) ) -> ( ( f ` 1o ) = ( f ` 2o ) -> 1o = 2o ) )
45	43, 44	mtoi 670	. . . . . . . 8 |- ( ( f : 3o -1-1-> A /\ ( 1o e. 3o /\ 2o e. 3o ) ) -> -. ( f ` 1o ) = ( f ` 2o ) )
46	14, 20, 45	mpanr12 439	. . . . . . 7 |- ( f : 3o -1-1-> A -> -. ( f ` 1o ) = ( f ` 2o ) )
47	46	neqned 2421	. . . . . 6 |- ( f : 3o -1-1-> A -> ( f ` 1o ) =/= ( f ` 2o ) )
48	47	adantl 277	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 1o ) =/= ( f ` 2o ) )
49		neeq2 2428	. . . . . . 7 |- ( z = ( f ` 2o ) -> ( ( f ` (/) ) =/= z <-> ( f ` (/) ) =/= ( f ` 2o ) ) )
50		neeq2 2428	. . . . . . 7 |- ( z = ( f ` 2o ) -> ( ( f ` 1o ) =/= z <-> ( f ` 1o ) =/= ( f ` 2o ) ) )
51	49, 50	3anbi23d 1352	. . . . . 6 |- ( z = ( f ` 2o ) -> ( ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) <-> ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= ( f ` 2o ) /\ ( f ` 1o ) =/= ( f ` 2o ) ) ) )
52	51	rspcev 2923	. . . . 5 |- ( ( ( f ` 2o ) e. A /\ ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= ( f ` 2o ) /\ ( f ` 1o ) =/= ( f ` 2o ) ) ) -> E. z e. A ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) )
53	22, 29, 36, 48, 52	syl13anc 1276	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> E. z e. A ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) )
54		neeq2 2428	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( ( f ` (/) ) =/= y <-> ( f ` (/) ) =/= ( f ` 1o ) ) )
55		biidd 172	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( ( f ` (/) ) =/= z <-> ( f ` (/) ) =/= z ) )
56		neeq1 2427	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( y =/= z <-> ( f ` 1o ) =/= z ) )
57	54, 55, 56	3anbi123d 1349	. . . . . 6 |- ( y = ( f ` 1o ) -> ( ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) <-> ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) ) )
58	57	rexbidv 2545	. . . . 5 |- ( y = ( f ` 1o ) -> ( E. z e. A ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) <-> E. z e. A ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) ) )
59	58	rspcev 2923	. . . 4 |- ( ( ( f ` 1o ) e. A /\ E. z e. A ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) ) -> E. y e. A E. z e. A ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) )
60	16, 53, 59	syl2anc 411	. . 3 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> E. y e. A E. z e. A ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) )
61		neeq1 2427	. . . . . 6 |- ( x = ( f ` (/) ) -> ( x =/= y <-> ( f ` (/) ) =/= y ) )
62		neeq1 2427	. . . . . 6 |- ( x = ( f ` (/) ) -> ( x =/= z <-> ( f ` (/) ) =/= z ) )
63		biidd 172	. . . . . 6 |- ( x = ( f ` (/) ) -> ( y =/= z <-> y =/= z ) )
64	61, 62, 63	3anbi123d 1349	. . . . 5 |- ( x = ( f ` (/) ) -> ( ( x =/= y /\ x =/= z /\ y =/= z ) <-> ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) ) )
65	64	2rexbidv 2569	. . . 4 |- ( x = ( f ` (/) ) -> ( E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) <-> E. y e. A E. z e. A ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) ) )
66	65	rspcev 2923	. . 3 |- ( ( ( f ` (/) ) e. A /\ E. y e. A E. z e. A ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) ) -> E. x e. A E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) )
67	10, 60, 66	syl2anc 411	. 2 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> E. x e. A E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) )
68	1, 67	exlimddv 1950	1 |- ( 3o ~<_ A -> E. x e. A E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   (/)c0 3512   class class class wbr 4114   Ord word 4488   suc csuc 4491   omcom 4717   -->wf 5353   -1-1->wf1 5354   ` cfv 5357   1oc1o 6653   2oc2o 6654   3oc3o 6655   ~<_ cdom 6987

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fv 5365  df-1o 6660  df-2o 6661  df-3o 6662  df-dom 6990

This theorem is referenced by:  pw1ndom3  16903