Theorem dford3 37595

Description: Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
dford3	|- ( Ord N <-> ( Tr N /\ A. x e. N Tr x ) )

Distinct variable group:   x, N

Proof of Theorem dford3

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtr 5737	. . 3 |- ( Ord N -> Tr N )
2		ordelord 5745	. . . . 5 |- ( ( Ord N /\ x e. N ) -> Ord x )
3		ordtr 5737	. . . . 5 |- ( Ord x -> Tr x )
4	2, 3	syl 17	. . . 4 |- ( ( Ord N /\ x e. N ) -> Tr x )
5	4	ralrimiva 2966	. . 3 |- ( Ord N -> A. x e. N Tr x )
6	1, 5	jca 554	. 2 |- ( Ord N -> ( Tr N /\ A. x e. N Tr x ) )
7		simpl 473	. . 3 |- ( ( Tr N /\ A. x e. N Tr x ) -> Tr N )
8		dford3lem1 37593	. . . . 5 |- ( ( Tr N /\ A. x e. N Tr x ) -> A. a e. N ( Tr a /\ A. x e. a Tr x ) )
9		dford3lem2 37594	. . . . . 6 |- ( ( Tr a /\ A. x e. a Tr x ) -> a e. On )
10	9	ralimi 2952	. . . . 5 |- ( A. a e. N ( Tr a /\ A. x e. a Tr x ) -> A. a e. N a e. On )
11	8, 10	syl 17	. . . 4 |- ( ( Tr N /\ A. x e. N Tr x ) -> A. a e. N a e. On )
12		dfss3 3592	. . . 4 |- ( N C_ On <-> A. a e. N a e. On )
13	11, 12	sylibr 224	. . 3 |- ( ( Tr N /\ A. x e. N Tr x ) -> N C_ On )
14		ordon 6982	. . . 4 |- Ord On
15	14	a1i 11	. . 3 |- ( ( Tr N /\ A. x e. N Tr x ) -> Ord On )
16		trssord 5740	. . 3 |- ( ( Tr N /\ N C_ On /\ Ord On ) -> Ord N )
17	7, 13, 15, 16	syl3anc 1326	. 2 |- ( ( Tr N /\ A. x e. N Tr x ) -> Ord N )
18	6, 17	impbii 199	1 |- ( Ord N <-> ( Tr N /\ A. x e. N Tr x ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   Tr wtr 4752   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  dford4  37596