Theorem onintonm 4261

Description: The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
onintonm	|- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )

Distinct variable group:   x, A

Proof of Theorem onintonm

Step	Hyp	Ref	Expression
1		ssel 2993	. . . . . . 7 |- ( A C_ On -> ( x e. A -> x e. On ) )
2		eloni 4130	. . . . . . . 8 |- ( x e. On -> Ord x )
3		ordtr 4133	. . . . . . . 8 |- ( Ord x -> Tr x )
4	2, 3	syl 14	. . . . . . 7 |- ( x e. On -> Tr x )
5	1, 4	syl6 33	. . . . . 6 |- ( A C_ On -> ( x e. A -> Tr x ) )
6	5	ralrimiv 2433	. . . . 5 |- ( A C_ On -> A. x e. A Tr x )
7		trint 3890	. . . . 5 |- ( A. x e. A Tr x -> Tr |^| A )
8	6, 7	syl 14	. . . 4 |- ( A C_ On -> Tr |^| A )
9	8	adantr 270	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> Tr |^| A )
10		nfv 1461	. . . . 5 |- F/ x A C_ On
11		nfe1 1425	. . . . 5 |- F/ x E. x x e. A
12	10, 11	nfan 1497	. . . 4 |- F/ x ( A C_ On /\ E. x x e. A )
13		intssuni2m 3660	. . . . . . . 8 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ U. On )
14		unon 4255	. . . . . . . 8 |- U. On = On
15	13, 14	syl6sseq 3045	. . . . . . 7 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ On )
16	15	sseld 2998	. . . . . 6 |- ( ( A C_ On /\ E. x x e. A ) -> ( x e. |^| A -> x e. On ) )
17	16, 2	syl6 33	. . . . 5 |- ( ( A C_ On /\ E. x x e. A ) -> ( x e. |^| A -> Ord x ) )
18	17, 3	syl6 33	. . . 4 |- ( ( A C_ On /\ E. x x e. A ) -> ( x e. |^| A -> Tr x ) )
19	12, 18	ralrimi 2432	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> A. x e. |^| A Tr x )
20		dford3 4122	. . 3 |- ( Ord |^| A <-> ( Tr |^| A /\ A. x e. |^| A Tr x ) )
21	9, 19, 20	sylanbrc 408	. 2 |- ( ( A C_ On /\ E. x x e. A ) -> Ord |^| A )
22		inteximm 3924	. . . 4 |- ( E. x x e. A -> |^| A e. _V )
23	22	adantl 271	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. _V )
24		elong 4128	. . 3 |- ( |^| A e. _V -> ( |^| A e. On <-> Ord |^| A ) )
25	23, 24	syl 14	. 2 |- ( ( A C_ On /\ E. x x e. A ) -> ( |^| A e. On <-> Ord |^| A ) )
26	21, 25	mpbird 165	1 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   A.wral 2348   _Vcvv 2601   C_ wss 2973   U.cuni 3601   |^|cint 3636   Tr wtr 3875   Ord word 4117   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by:  onintrab2im  4262