Theorem 3dom 16608

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 6920	. 2 |- ( 3o ~<_ A -> E. f f : 3o -1-1-> A )
2		f1f 5542	. . . . 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 6609	. . . . . . 7 |- (/) e. 2o
5		elelsuc 4506	. . . . . . 7 |- ( (/) e. 2o -> (/) e. suc 2o )
6	4, 5	ax-mp 5	. . . . . 6 |- (/) e. suc 2o
7		df-3o 6584	. . . . . 6 |- 3o = suc 2o
8	6, 7	eleqtrri 2307	. . . . 5 |- (/) e. 3o
9	8	a1i 9	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> (/) e. 3o )
10	3, 9	ffvelcdmd 5783	. . 3 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` (/) ) e. A )
11		1lt2o 6610	. . . . . . . 8 |- 1o e. 2o
12		elelsuc 4506	. . . . . . . 8 |- ( 1o e. 2o -> 1o e. suc 2o )
13	11, 12	ax-mp 5	. . . . . . 7 |- 1o e. suc 2o
14	13, 7	eleqtrri 2307	. . . . . 6 |- 1o e. 3o
15	14	a1i 9	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> 1o e. 3o )
16	3, 15	ffvelcdmd 5783	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 1o ) e. A )
17		2onn 6689	. . . . . . . . . 10 |- 2o e. om
18	17	elexi 2815	. . . . . . . . 9 |- 2o e. _V
19	18	sucid 4514	. . . . . . . 8 |- 2o e. suc 2o
20	19, 7	eleqtrri 2307	. . . . . . 7 |- 2o e. 3o
21	20	a1i 9	. . . . . 6 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> 2o e. 3o )
22	3, 21	ffvelcdmd 5783	. . . . 5 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> ( f ` 2o ) e. A )
23		1n0 6600	. . . . . . . . . 10 |- 1o =/= (/)
24	23	nesymi 2448	. . . . . . . . 9 |- -. (/) = 1o
25		f1veqaeq 5910	. . . . . . . . 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 2409	. . . . . 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 6592	. . . . . . . . . 10 |- 2o =/= (/)
31	30	nesymi 2448	. . . . . . . . 9 |- -. (/) = 2o
32		f1veqaeq 5910	. . . . . . . . 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 2409	. . . . . 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 4710	. . . . . . . . . . . . 13 |- ( 2o e. om -> Ord 2o )
38	17, 37	ax-mp 5	. . . . . . . . . . . 12 |- Ord 2o
39		ordirr 4640	. . . . . . . . . . . 12 |- ( Ord 2o -> -. 2o e. 2o )
40	38, 39	ax-mp 5	. . . . . . . . . . 11 |- -. 2o e. 2o
41		eleq1 2294	. . . . . . . . . . 11 |- ( 1o = 2o -> ( 1o e. 2o <-> 2o e. 2o ) )
42	40, 41	mtbiri 681	. . . . . . . . . 10 |- ( 1o = 2o -> -. 1o e. 2o )
43	11, 42	mt2 645	. . . . . . . . 9 |- -. 1o = 2o
44		f1veqaeq 5910	. . . . . . . . 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 2409	. . . . . 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 2416	. . . . . . 7 |- ( z = ( f ` 2o ) -> ( ( f ` (/) ) =/= z <-> ( f ` (/) ) =/= ( f ` 2o ) ) )
50		neeq2 2416	. . . . . . 7 |- ( z = ( f ` 2o ) -> ( ( f ` 1o ) =/= z <-> ( f ` 1o ) =/= ( f ` 2o ) ) )
51	49, 50	3anbi23d 1351	. . . . . 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 2910	. . . . 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 1275	. . . 4 |- ( ( 3o ~<_ A /\ f : 3o -1-1-> A ) -> E. z e. A ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) )
54		neeq2 2416	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( ( f ` (/) ) =/= y <-> ( f ` (/) ) =/= ( f ` 1o ) ) )
55		biidd 172	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( ( f ` (/) ) =/= z <-> ( f ` (/) ) =/= z ) )
56		neeq1 2415	. . . . . . 7 |- ( y = ( f ` 1o ) -> ( y =/= z <-> ( f ` 1o ) =/= z ) )
57	54, 55, 56	3anbi123d 1348	. . . . . 6 |- ( y = ( f ` 1o ) -> ( ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) <-> ( ( f ` (/) ) =/= ( f ` 1o ) /\ ( f ` (/) ) =/= z /\ ( f ` 1o ) =/= z ) ) )
58	57	rexbidv 2533	. . . . 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 2910	. . . 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 2415	. . . . . 6 |- ( x = ( f ` (/) ) -> ( x =/= y <-> ( f ` (/) ) =/= y ) )
62		neeq1 2415	. . . . . 6 |- ( x = ( f ` (/) ) -> ( x =/= z <-> ( f ` (/) ) =/= z ) )
63		biidd 172	. . . . . 6 |- ( x = ( f ` (/) ) -> ( y =/= z <-> y =/= z ) )
64	61, 62, 63	3anbi123d 1348	. . . . 5 |- ( x = ( f ` (/) ) -> ( ( x =/= y /\ x =/= z /\ y =/= z ) <-> ( ( f ` (/) ) =/= y /\ ( f ` (/) ) =/= z /\ y =/= z ) ) )
65	64	2rexbidv 2557	. . . 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 2910	. . 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 1947	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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   (/)c0 3494   class class class wbr 4088   Ord word 4459   suc csuc 4462   omcom 4688   -->wf 5322   -1-1->wf1 5323   ` cfv 5326   1oc1o 6575   2oc2o 6576   3oc3o 6577   ~<_ cdom 6908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334  df-1o 6582  df-2o 6583  df-3o 6584  df-dom 6911

This theorem is referenced by:  pw1ndom3  16610