Theorem onsucwordi 43277

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 6395	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6395	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordsucsssuc 7842	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
4	1, 2, 3	syl2an 596	. 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 2105   ⊆ wss 3962  Ord word 6384  Oncon0 6385  suc csuc 6387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-suc 6391

This theorem is referenced by:  onsucunipr  43361