Theorem ordsucss 4196

Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)

Assertion

Ref	Expression
ordsucss	⊢ (Ord B → (A ∈ B → suc A ⊆ B))

Proof of Theorem ordsucss

Step	Hyp	Ref	Expression
1		ordtr 4081	. 2 ⊢ (Ord B → Tr B)
2		trss 3854	. . . . 5 ⊢ (Tr B → (A ∈ B → A ⊆ B))
3		snssi 3499	. . . . . 6 ⊢ (A ∈ B → {A} ⊆ B)
4	3	a1i 9	. . . . 5 ⊢ (Tr B → (A ∈ B → {A} ⊆ B))
5	2, 4	jcad 291	. . . 4 ⊢ (Tr B → (A ∈ B → (A ⊆ B ∧ {A} ⊆ B)))
6		unss 3111	. . . 4 ⊢ ((A ⊆ B ∧ {A} ⊆ B) ↔ (A ∪ {A}) ⊆ B)
7	5, 6	syl6ib 150	. . 3 ⊢ (Tr B → (A ∈ B → (A ∪ {A}) ⊆ B))
8		df-suc 4074	. . . 4 ⊢ suc A = (A ∪ {A})
9	8	sseq1i 2963	. . 3 ⊢ (suc A ⊆ B ↔ (A ∪ {A}) ⊆ B)
10	7, 9	syl6ibr 151	. 2 ⊢ (Tr B → (A ∈ B → suc A ⊆ B))
11	1, 10	syl 14	1 ⊢ (Ord B → (A ∈ B → suc A ⊆ B))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  {csn 3367  Tr wtr 3845  Ord word 4065  suc csuc 4068

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074

This theorem is referenced by:  ordelsuc  4197  tfrlemibfn  5883  sucinc2  5965  prarloclemn  6482