Theorem 3dom 16381

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 6906	. 2 |- ( 3o ~<_ A -> E. f f : 3o -1-1-> A )
2		f1f 5533	. . . . 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 6595	. . . . . . 7 |- (/) e. 2o
5		elelsuc 4500	. . . . . . 7 |- ( (/) e. 2o -> (/) e. suc 2o )
6	4, 5	ax-mp 5	. . . . . 6 |- (/) e. suc 2o
7		df-3o 6570	. . . . . 6 |- 3o = suc 2o
8	6, 7	eleqtrri 2305	. . . . 5 |- (/) e. 3o
9	8	a1i 9	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> (/) e. 3o )
10	3, 9	ffvelcdmd 5773	. . 3 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` (/) ) e. A )
11		1lt2o 6596	. . . . . . . 8 |- 1o e. 2o
12		elelsuc 4500	. . . . . . . 8 |- ( 1o e. 2o -> 1o e. suc 2o )
13	11, 12	ax-mp 5	. . . . . . 7 |- 1o e. suc 2o
14	13, 7	eleqtrri 2305	. . . . . 6 |- 1o e. 3o
15	14	a1i 9	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> 1o e. 3o )
16	3, 15	ffvelcdmd 5773	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 1o ) e. A )
17		2onn 6675	. . . . . . . . . 10 |- 2o e. om
18	17	elexi 2812	. . . . . . . . 9 |- 2o e. _V
19	18	sucid 4508	. . . . . . . 8 |- 2o e. suc 2o
20	19, 7	eleqtrri 2305	. . . . . . 7 |- 2o e. 3o
21	20	a1i 9	. . . . . 6 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> 2o e. 3o )
22	3, 21	ffvelcdmd 5773	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 2o ) e. A )
23		1n0 6586	. . . . . . . . . 10 |- 1o =/= (/)
24	23	nesymi 2446	. . . . . . . . 9 |- -. (/) = 1o
25		f1veqaeq 5899	. . . . . . . . 9 |- ( ( f : 3o -1-1-> A /\ ( (/) e. 3o /\ 1o e. 3o ) ) -> ( ( f ` (/) ) = ( f ` 1o ) -> (/) = 1o ) )
26	24, 25	mtoi 668	. . . . . . . 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 2407	. . . . . 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 6578	. . . . . . . . . 10 |- 2o =/= (/)
31	30	nesymi 2446	. . . . . . . . 9 |- -. (/) = 2o
32		f1veqaeq 5899	. . . . . . . . 9 |- ( ( f : 3o -1-1-> A /\ ( (/) e. 3o /\ 2o e. 3o ) ) -> ( ( f ` (/) ) = ( f ` 2o ) -> (/) = 2o ) )
33	31, 32	mtoi 668	. . . . . . . 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 2407	. . . . . 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 4704	. . . . . . . . . . . . 13 |- ( 2o e. om -> Ord 2o )
38	17, 37	ax-mp 5	. . . . . . . . . . . 12 |- Ord 2o
39		ordirr 4634	. . . . . . . . . . . 12 |- ( Ord 2o -> -. 2o e. 2o )
40	38, 39	ax-mp 5	. . . . . . . . . . 11 |- -. 2o e. 2o
41		eleq1 2292	. . . . . . . . . . 11 |- ( 1o = 2o -> ( 1o e. 2o <-> 2o e. 2o ) )
42	40, 41	mtbiri 679	. . . . . . . . . 10 |- ( 1o = 2o -> -. 1o e. 2o )
43	11, 42	mt2 643	. . . . . . . . 9 |- -. 1o = 2o
44		f1veqaeq 5899	. . . . . . . . 9 |- ( ( f : 3o -1-1-> A /\ ( 1o e. 3o /\ 2o e. 3o ) ) -> ( ( f ` 1o ) = ( f ` 2o ) -> 1o = 2o ) )
45	43, 44	mtoi 668	. . . . . . . 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 2407	. . . . . 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 2414	. . . . . . 7 |- ( z = ( f ` 2o ) -> ( ( f ` (/) ) =/= z <-> ( f ` (/) ) =/= ( f ` 2o ) ) )
50		neeq2 2414	. . . . . . 7 |- ( z = ( f ` 2o ) -> ( ( f ` 1o ) =/= z <-> ( f ` 1o ) =/= ( f ` 2o ) ) )
51	49, 50	3anbi23d 1349	. . . . . 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 2907	. . . . 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 1273	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> E. z e. A ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) )
54		neeq2 2414	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( ( f ` (/) ) =/= y <-> ( f ` (/) ) =/= ( f ` 1o ) ) )
55		biidd 172	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( ( f ` (/) ) =/= z <-> ( f ` (/) ) =/= z ) )
56		neeq1 2413	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( y =/= z <-> ( f ` 1o ) =/= z ) )
57	54, 55, 56	3anbi123d 1346	. . . . . 6 |- ( y = ( f ` 1o ) -> ( ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) <-> ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) ) )
58	57	rexbidv 2531	. . . . 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 2907	. . . 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 2413	. . . . . 6 |- ( x = ( f ` (/) ) -> ( x =/= y <-> ( f ` (/) ) =/= y ) )
62		neeq1 2413	. . . . . 6 |- ( x = ( f ` (/) ) -> ( x =/= z <-> ( f ` (/) ) =/= z ) )
63		biidd 172	. . . . . 6 |- ( x = ( f ` (/) ) -> ( y =/= z <-> y =/= z ) )
64	61, 62, 63	3anbi123d 1346	. . . . 5 |- ( x = ( f ` (/) ) -> ( ( x =/= y /\ x =/= z /\ y =/= z ) <-> ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) ) )
65	64	2rexbidv 2555	. . . 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 2907	. . 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 1945	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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   (/)c0 3491   class class class wbr 4083   Ord word 4453   suc csuc 4456   omcom 4682   -->wf 5314   -1-1->wf1 5315   ` cfv 5318   1oc1o 6561   2oc2o 6562   3oc3o 6563   ~<_ cdom 6894

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fv 5326  df-1o 6568  df-2o 6569  df-3o 6570  df-dom 6897

This theorem is referenced by:  pw1ndom3  16383