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Theorem ordsucun 7524
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 6264 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordsuc 7513 . . . . 5 (Ord (𝐴𝐵) ↔ Ord suc (𝐴𝐵))
3 ordelon 6187 . . . . . 6 ((Ord suc (𝐴𝐵) ∧ 𝑥 ∈ suc (𝐴𝐵)) → 𝑥 ∈ On)
43ex 416 . . . . 5 (Ord suc (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
52, 4sylbi 220 . . . 4 (Ord (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
61, 5syl 17 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
7 ordsuc 7513 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
8 ordsuc 7513 . . . 4 (Ord 𝐵 ↔ Ord suc 𝐵)
9 ordun 6264 . . . . 5 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10 ordelon 6187 . . . . . 6 ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
1110ex 416 . . . . 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 600 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14 ordssun 6262 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
1514adantl 485 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
16 ordsssuc 6249 . . . . . . 7 ((𝑥 ∈ On ∧ Ord (𝐴𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
171, 16sylan2 595 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
18 ordsssuc 6249 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
1918adantrr 716 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐴𝑥 ∈ suc 𝐴))
20 ordsssuc 6249 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2120adantrl 715 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2219, 21orbi12d 916 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵)))
2315, 17, 223bitr3d 312 . . . . 5 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵)))
24 elun 4079 . . . . 5 (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵))
2523, 24syl6bbr 292 . . . 4 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
2625expcom 417 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
276, 13, 26pm5.21ndd 384 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
2827eqrdv 2799 1 ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  cun 3882  wss 3884  Ord word 6162  Oncon0 6163  suc csuc 6165
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167  df-suc 6169
This theorem is referenced by:  rankprb  9268  noetalem4  33334
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