Theorem onsucsssucr 4172

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4184. (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 4164	. . 3 ⊢ (Ord B → Ord suc B)
2		ordelsuc 4169	. . 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 4053	. . . 4 ⊢ (Ord B → Tr B)
5		trsucss 4098	. . . 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 1366   ⊆ wss 2885  Tr wtr 3817  Ord word 4037  Oncon0 4038  suc csuc 4040

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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995

This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-uni 3544  df-tr 3818  df-iord 4041  df-suc 4046

This theorem is referenced by:  nnsucsssuc  5974