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Theorem orddif0suc 43258
Description: For any distinct pair of ordinals, if the set difference between the greater and the successor of the lesser is empty, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 17-Jan-2025.)
Assertion
Ref Expression
orddif0suc ((𝐴𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))

Proof of Theorem orddif0suc
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . . . 8 ((𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐵)
2 ordelon 6410 . . . . . . . . 9 ((Ord 𝐵𝐴𝐵) → 𝐴 ∈ On)
32ancoms 458 . . . . . . . 8 ((𝐴𝐵 ∧ Ord 𝐵) → 𝐴 ∈ On)
4 ordeldifsucon 43249 . . . . . . . 8 ((Ord 𝐵𝐴 ∈ On) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐𝐵𝐴𝑐)))
51, 3, 4syl2anc 584 . . . . . . 7 ((𝐴𝐵 ∧ Ord 𝐵) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐𝐵𝐴𝑐)))
65biancomd 463 . . . . . 6 ((𝐴𝐵 ∧ Ord 𝐵) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝐴𝑐𝑐𝐵)))
7 ordelon 6410 . . . . . . . . . 10 ((Ord 𝐵𝑐𝐵) → 𝑐 ∈ On)
87ad2ant2l 746 . . . . . . . . 9 (((𝐴𝐵 ∧ Ord 𝐵) ∧ (𝐴𝑐𝑐𝐵)) → 𝑐 ∈ On)
98ex 412 . . . . . . . 8 ((𝐴𝐵 ∧ Ord 𝐵) → ((𝐴𝑐𝑐𝐵) → 𝑐 ∈ On))
109pm4.71rd 562 . . . . . . 7 ((𝐴𝐵 ∧ Ord 𝐵) → ((𝐴𝑐𝑐𝐵) ↔ (𝑐 ∈ On ∧ (𝐴𝑐𝑐𝐵))))
11 df-an 396 . . . . . . 7 ((𝑐 ∈ On ∧ (𝐴𝑐𝑐𝐵)) ↔ ¬ (𝑐 ∈ On → ¬ (𝐴𝑐𝑐𝐵)))
1210, 11bitrdi 287 . . . . . 6 ((𝐴𝐵 ∧ Ord 𝐵) → ((𝐴𝑐𝑐𝐵) ↔ ¬ (𝑐 ∈ On → ¬ (𝐴𝑐𝑐𝐵))))
136, 12bitr2d 280 . . . . 5 ((𝐴𝐵 ∧ Ord 𝐵) → (¬ (𝑐 ∈ On → ¬ (𝐴𝑐𝑐𝐵)) ↔ 𝑐 ∈ (𝐵 ∖ suc 𝐴)))
1413con1bid 355 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → (¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ On → ¬ (𝐴𝑐𝑐𝐵))))
1514albidv 1918 . . 3 ((𝐴𝐵 ∧ Ord 𝐵) → (∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴𝑐𝑐𝐵))))
16 eq0 4356 . . 3 ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴))
17 df-ral 3060 . . 3 (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴𝑐𝑐𝐵)))
1815, 16, 173bitr4g 314 . 2 ((𝐴𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵)))
19 ordnexbtwnsuc 43257 . 2 ((𝐴𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴𝑐𝑐𝐵) → 𝐵 = suc 𝐴))
2018, 19sylbid 240 1 ((𝐴𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059  cdif 3960  c0 4339  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by: (None)
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