Theorem onintonm 4428

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 3086	. . . . . . 7 |- ( A C_ On -> ( x e. A -> x e. On ) )
2		eloni 4292	. . . . . . . 8 |- ( x e. On -> Ord x )
3		ordtr 4295	. . . . . . . 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 2502	. . . . 5 |- ( A C_ On -> A. x e. A Tr x )
7		trint 4036	. . . . 5 |- ( A. x e. A Tr x -> Tr |^| A )
8	6, 7	syl 14	. . . 4 |- ( A C_ On -> Tr |^| A )
9	8	adantr 274	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> Tr |^| A )
10		nfv 1508	. . . . 5 |- F/ x A C_ On
11		nfe1 1472	. . . . 5 |- F/ x E. x x e. A
12	10, 11	nfan 1544	. . . 4 |- F/ x ( A C_ On /\ E. x x e. A )
13		intssuni2m 3790	. . . . . . . 8 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ U. On )
14		unon 4422	. . . . . . . 8 |- U. On = On
15	13, 14	sseqtrdi 3140	. . . . . . 7 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ On )
16	15	sseld 3091	. . . . . 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 2501	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> A. x e. |^| A Tr x )
20		dford3 4284	. . 3 |- ( Ord |^| A <-> ( Tr |^| A /\ A. x e. |^| A Tr x ) )
21	9, 19, 20	sylanbrc 413	. 2 |- ( ( A C_ On /\ E. x x e. A ) -> Ord |^| A )
22		inteximm 4069	. . . 4 |- ( E. x x e. A -> |^| A e. _V )
23	22	adantl 275	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. _V )
24		elong 4290	. . 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 166	1 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1468   e. wcel 1480   A.wral 2414   _Vcvv 2681   C_ wss 3066   U.cuni 3731   |^|cint 3766   Tr wtr 4021   Ord word 4279   Oncon0 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288

This theorem is referenced by:  onintrab2im  4429