Theorem onsucsssucr 4182

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4194. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)

Assertion

Ref	Expression
onsucsssucr	⊢ ((A ∈ On ∧ Ord B) → (suc A ⊆ suc B → A ⊆ B))

Proof of Theorem onsucsssucr

Step	Hyp	Ref	Expression
1		ordsucim 4174	. . 3 ⊢ (Ord B → Ord suc B)
2		ordelsuc 4179	. . 3 ⊢ ((A ∈ On ∧ Ord suc B) → (A ∈ suc B ↔ suc A ⊆ suc B))
3	1, 2	sylan2 270	. 2 ⊢ ((A ∈ On ∧ Ord B) → (A ∈ suc B ↔ suc A ⊆ suc B))
4		ordtr 4062	. . . 4 ⊢ (Ord B → Tr B)
5		trsucss 4108	. . . 4 ⊢ (Tr B → (A ∈ suc B → A ⊆ B))
6	4, 5	syl 14	. . 3 ⊢ (Ord B → (A ∈ suc B → A ⊆ B))
7	6	adantl 262	. 2 ⊢ ((A ∈ On ∧ Ord B) → (A ∈ suc B → A ⊆ B))
8	3, 7	sylbird 159	1 ⊢ ((A ∈ On ∧ Ord B) → (suc A ⊆ suc B → A ⊆ B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1374   ⊆ wss 2892  Tr wtr 3826  Ord word 4046  Oncon0 4047  suc csuc 4049

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2897  df-in 2899  df-ss 2906  df-sn 3354  df-uni 3553  df-tr 3827  df-iord 4050  df-suc 4055

This theorem is referenced by:  nnsucsssuc  5984