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Theorem ordsucun 7845
Description: The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordsucun ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))

Proof of Theorem ordsucun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordun 6488 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordsuc 7833 . . . . 5 (Ord (𝐴𝐵) ↔ Ord suc (𝐴𝐵))
3 ordelon 6408 . . . . . 6 ((Ord suc (𝐴𝐵) ∧ 𝑥 ∈ suc (𝐴𝐵)) → 𝑥 ∈ On)
43ex 412 . . . . 5 (Ord suc (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
52, 4sylbi 217 . . . 4 (Ord (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
61, 5syl 17 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
7 ordsuc 7833 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
8 ordsuc 7833 . . . 4 (Ord 𝐵 ↔ Ord suc 𝐵)
9 ordun 6488 . . . . 5 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10 ordelon 6408 . . . . . 6 ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
1110ex 412 . . . . 5 (Ord (suc 𝐴 ∪ suc 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
129, 11syl 17 . . . 4 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
137, 8, 12syl2anb 598 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14 ordssun 6486 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
1514adantl 481 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
16 ordsssuc 6473 . . . . . . 7 ((𝑥 ∈ On ∧ Ord (𝐴𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
171, 16sylan2 593 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
18 ordsssuc 6473 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
1918adantrr 717 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐴𝑥 ∈ suc 𝐴))
20 ordsssuc 6473 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2120adantrl 716 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2219, 21orbi12d 919 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵)))
2315, 17, 223bitr3d 309 . . . . 5 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵)))
24 elun 4153 . . . . 5 (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵))
2523, 24bitr4di 289 . . . 4 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
2625expcom 413 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
276, 13, 26pm5.21ndd 379 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
2827eqrdv 2735 1 ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  cun 3949  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
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:  rankprb  9891
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