Theorem onintonm 4243

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 2939	. . . . . . 7 |- ( A C_ On -> ( x e. A -> x e. On ) )
2		eloni 4112	. . . . . . . 8 |- ( x e. On -> Ord x )
3		ordtr 4115	. . . . . . . 8 |- ( Ord x -> Tr x )
4	2, 3	syl 14	. . . . . . 7 |- ( x e. On -> Tr x )
5	1, 4	syl6 29	. . . . . 6 |- ( A C_ On -> ( x e. A -> Tr x ) )
6	5	ralrimiv 2391	. . . . 5 |- ( A C_ On -> A. x e. A Tr x )
7		trint 3869	. . . . 5 |- ( A. x e. A Tr x -> Tr |^| A )
8	6, 7	syl 14	. . . 4 |- ( A C_ On -> Tr |^| A )
9	8	adantr 261	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> Tr |^| A )
10		nfv 1421	. . . . 5 |- F/ x A C_ On
11		nfe1 1385	. . . . 5 |- F/ x E. x x e. A
12	10, 11	nfan 1457	. . . 4 |- F/ x ( A C_ On /\ E. x x e. A )
13		intssuni2m 3639	. . . . . . . 8 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ U. On )
14		unon 4237	. . . . . . . 8 |- U. On = On
15	13, 14	syl6sseq 2991	. . . . . . 7 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ On )
16	15	sseld 2944	. . . . . 6 |- ( ( A C_ On /\ E. x x e. A ) -> ( x e. |^| A -> x e. On ) )
17	16, 2	syl6 29	. . . . 5 |- ( ( A C_ On /\ E. x x e. A ) -> ( x e. |^| A -> Ord x ) )
18	17, 3	syl6 29	. . . 4 |- ( ( A C_ On /\ E. x x e. A ) -> ( x e. |^| A -> Tr x ) )
19	12, 18	ralrimi 2390	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> A. x e. |^| A Tr x )
20		dford3 4104	. . 3 |- ( Ord |^| A <-> ( Tr |^| A /\ A. x e. |^| A Tr x ) )
21	9, 19, 20	sylanbrc 394	. 2 |- ( ( A C_ On /\ E. x x e. A ) -> Ord |^| A )
22		inteximm 3903	. . . 4 |- ( E. x x e. A -> |^| A e. _V )
23	22	adantl 262	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. _V )
24		elong 4110	. . 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 156	1 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   A.wral 2306   _Vcvv 2557   C_ wss 2917   U.cuni 3580   |^|cint 3615   Tr wtr 3854   Ord word 4099   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by:  onintrab2im  4244