Theorem onsucsssucr 4420

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4437. (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 4411	. . 3 ⊢ (Ord 𝐵 → Ord suc 𝐵)
2		ordelsuc 4416	. . 3 ⊢ ((𝐴 ∈ On ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
3	1, 2	sylan2 284	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
4		ordtr 4295	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
5		trsucss 4340	. . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
6	4, 5	syl 14	. . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
7	6	adantl 275	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
8	3, 7	sylbird 169	1 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1480   ⊆ wss 3066  Tr wtr 4021  Ord word 4279  Oncon0 4280  suc csuc 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-uni 3732  df-tr 4022  df-iord 4283  df-suc 4288

This theorem is referenced by:  nnsucsssuc  6381