Theorem onintonm 4269

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 2994	. . . . . . 7 |- ( A C_ On -> ( x e. A -> x e. On ) )
2		eloni 4138	. . . . . . . 8 |- ( x e. On -> Ord x )
3		ordtr 4141	. . . . . . . 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 2434	. . . . 5 |- ( A C_ On -> A. x e. A Tr x )
7		trint 3898	. . . . 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 1462	. . . . 5 |- F/ x A C_ On
11		nfe1 1426	. . . . 5 |- F/ x E. x x e. A
12	10, 11	nfan 1498	. . . 4 |- F/ x ( A C_ On /\ E. x x e. A )
13		intssuni2m 3668	. . . . . . . 8 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ U. On )
14		unon 4263	. . . . . . . 8 |- U. On = On
15	13, 14	syl6sseq 3046	. . . . . . 7 |- ( ( A C_ On /\ E. x x e. A ) -> |^| A C_ On )
16	15	sseld 2999	. . . . . 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 2433	. . 3 |- ( ( A C_ On /\ E. x x e. A ) -> A. x e. |^| A Tr x )
20		dford3 4130	. . 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 3932	. . . 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 4136	. . 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 1422   e. wcel 1434   A.wral 2349   _Vcvv 2602   C_ wss 2974   U.cuni 3609   |^|cint 3644   Tr wtr 3883   Ord word 4125   Oncon0 4126

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134

This theorem is referenced by:  onintrab2im  4270