Theorem ordsucsssuc 6537

Description: The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)

Assertion

Ref	Expression
ordsucsssuc	|- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> suc A C_ suc B ) )

Proof of Theorem ordsucsssuc

Step	Hyp	Ref	Expression
1		ordsucelsuc 6536	. . . 4 |- ( Ord A -> ( B e. A <-> suc B e. suc A ) )
2	1	notbid 294	. . 3 |- ( Ord A -> ( -. B e. A <-> -. suc B e. suc A ) )
3	2	adantr 465	. 2 |- ( ( Ord A /\ Ord B ) -> ( -. B e. A <-> -. suc B e. suc A ) )
4		ordtri1 4853	. 2 |- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> -. B e. A ) )
5		ordsuc 6528	. . 3 |- ( Ord A <-> Ord suc A )
6		ordsuc 6528	. . 3 |- ( Ord B <-> Ord suc B )
7		ordtri1 4853	. . 3 |- ( ( Ord suc A /\ Ord suc B ) -> ( suc A C_ suc B <-> -. suc B e. suc A ) )
8	5, 6, 7	syl2anb 479	. 2 |- ( ( Ord A /\ Ord B ) -> ( suc A C_ suc B <-> -. suc B e. suc A ) )
9	3, 4, 8	3bitr4d 285	1 |- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> suc A C_ suc B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1758   C_ wss 3429   Ord word 4819   suc csuc 4822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826

This theorem is referenced by:  oawordri  7092  oeworde  7135  nnawordi  7163  bndrank  8152  rankmapu  8189  ackbij1b  8512  onsuct0  28424