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Theorem ordsucun 7765
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 6426 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordsuc 7753 . . . . 5 (Ord (𝐴𝐵) ↔ Ord suc (𝐴𝐵))
3 ordelon 6346 . . . . . 6 ((Ord suc (𝐴𝐵) ∧ 𝑥 ∈ suc (𝐴𝐵)) → 𝑥 ∈ On)
43ex 414 . . . . 5 (Ord suc (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
52, 4sylbi 216 . . . 4 (Ord (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
61, 5syl 17 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
7 ordsuc 7753 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
8 ordsuc 7753 . . . 4 (Ord 𝐵 ↔ Ord suc 𝐵)
9 ordun 6426 . . . . 5 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10 ordelon 6346 . . . . . 6 ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
1110ex 414 . . . . 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 599 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14 ordssun 6424 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
1514adantl 483 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
16 ordsssuc 6411 . . . . . . 7 ((𝑥 ∈ On ∧ Ord (𝐴𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
171, 16sylan2 594 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
18 ordsssuc 6411 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
1918adantrr 716 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐴𝑥 ∈ suc 𝐴))
20 ordsssuc 6411 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2120adantrl 715 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2219, 21orbi12d 918 . . . . . 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 4113 . . . . 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 415 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
276, 13, 26pm5.21ndd 381 . 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 205  wa 397  wo 846   = wceq 1542  wcel 2107  cun 3913  wss 3915  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  rankprb  9794
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