Theorem onintonm 4516

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 3149	. . . . . . 7 |- ( A C_ On -> ( x e. A -> x e. On ) )
2		eloni 4375	. . . . . . . 8 |- ( x e. On -> Ord x )
3		ordtr 4378	. . . . . . . 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 2549	. . . . 5 |- ( A C_ On -> A. x e. A Tr x )
7		trint 4116	. . . . 5 |- ( A. x e. A Tr x -> Tr |^| A )
8	6, 7	syl 14	. . . 4 |- ( A C_ On -> Tr |^| A )
9	8	adantr 276	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> Tr |^| A )
10		nfv 1528	. . . . 5 |- F/ x A C_ On
11		nfe1 1496	. . . . 5 |- F/ x E. x x e. A
12	10, 11	nfan 1565	. . . 4 |- F/ x ( A C_ On /\ E. x x e. A )
13		intssuni2m 3868	. . . . . . . 8 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ U. On )
14		unon 4510	. . . . . . . 8 |- U. On = On
15	13, 14	sseqtrdi 3203	. . . . . . 7 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ On )
16	15	sseld 3154	. . . . . 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 2548	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> A. x e. |^| A Tr x )
20		dford3 4367	. . 3 |- ( Ord |^| A <-> ( Tr |^| A /\ A. x e. |^| A Tr x ) )
21	9, 19, 20	sylanbrc 417	. 2 |- ( ( A C_ On /\ E. x x e. A ) -> Ord |^| A )
22		inteximm 4149	. . . 4 |- ( E. x x e. A -> |^| A e. _V )
23	22	adantl 277	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. _V )
24		elong 4373	. . 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 167	1 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1492   e. wcel 2148   A.wral 2455   _Vcvv 2737   C_ wss 3129   U.cuni 3809   |^|cint 3844   Tr wtr 4101   Ord word 4362   Oncon0 4363

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371

This theorem is referenced by:  onintrab2im  4517