Theorem onsucsssucr 4200

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4212. (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 4192	. . 3 ⊢ (Ord B → Ord suc B)
2		ordelsuc 4197	. . 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 4081	. . . 4 ⊢ (Ord B → Tr B)
5		trsucss 4126	. . . 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 1390   ⊆ wss 2911  Tr wtr 3845  Ord word 4065  Oncon0 4066  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-rex 2306  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:  nnsucsssuc  6010