Theorem onsucsssucr 4354

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4371. (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 4345	. . 3 ⊢ (Ord 𝐵 → Ord suc 𝐵)
2		ordelsuc 4350	. . 3 ⊢ ((𝐴 ∈ On ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
3	1, 2	sylan2 281	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
4		ordtr 4229	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
5		trsucss 4274	. . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
6	4, 5	syl 14	. . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
7	6	adantl 272	. 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 1445   ⊆ wss 3013  Tr wtr 3958  Ord word 4213  Oncon0 4214  suc csuc 4216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-suc 4222

This theorem is referenced by:  nnsucsssuc  6293