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Theorem onsucwordi 43301
Description: The successor operation preserves the less-than-or-equal relationship between ordinals. Lemma 3.1 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.)
Assertion
Ref Expression
onsucwordi ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))

Proof of Theorem onsucwordi
StepHypRef Expression
1 eloni 6394 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6394 . . 3 (𝐵 ∈ On → Ord 𝐵)
3 ordsucsssuc 7843 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
41, 2, 3syl2an 596 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
54biimpd 229 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wss 3951  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  onsucunipr  43385
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