Theorem onsucwordi 43597

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) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))

Proof of Theorem onsucwordi

Step	Hyp	Ref	Expression
1		eloni 6328	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6328	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordsucsssuc 7767	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
5	4	biimpd 229	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ⊆ wss 3902  Ord word 6317  Oncon0 6318  suc csuc 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-suc 6324

This theorem is referenced by:  onsucunipr  43681