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Theorem ordsssucb 43353
Description: An ordinal number is less than or equal to the successor of an ordinal class iff the ordinal number is either less than or equal to the ordinal class or the ordinal number is equal to the successor of the ordinal class. See also ordsssucim 43420, limsssuc 7872. (Contributed by RP, 22-Feb-2025.)
Assertion
Ref Expression
ordsssucb ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴𝐵𝐴 = suc 𝐵)))

Proof of Theorem ordsssucb
StepHypRef Expression
1 sspss 4101 . 2 (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵𝐴 = suc 𝐵))
2 ordsssuc 6472 . . . 4 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))
3 eloni 6393 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
4 ordsuci 7829 . . . . 5 (Ord 𝐵 → Ord suc 𝐵)
5 ordelpss 6411 . . . . 5 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵𝐴 ⊊ suc 𝐵))
63, 4, 5syl2an 596 . . . 4 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵𝐴 ⊊ suc 𝐵))
72, 6bitrd 279 . . 3 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ⊊ suc 𝐵))
87orbi1d 916 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = suc 𝐵) ↔ (𝐴 ⊊ suc 𝐵𝐴 = suc 𝐵)))
91, 8bitr4id 290 1 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴𝐵𝐴 = suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1539  wcel 2107  wss 3950  wpss 3951  Ord word 6382  Oncon0 6383  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389
This theorem is referenced by: (None)
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