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Theorem ordsucun 7172
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 5972 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
2 ordsuc 7161 . . . . 5 (Ord (𝐴𝐵) ↔ Ord suc (𝐴𝐵))
3 ordelon 5890 . . . . . 6 ((Ord suc (𝐴𝐵) ∧ 𝑥 ∈ suc (𝐴𝐵)) → 𝑥 ∈ On)
43ex 397 . . . . 5 (Ord suc (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
52, 4sylbi 207 . . . 4 (Ord (𝐴𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
61, 5syl 17 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) → 𝑥 ∈ On))
7 ordsuc 7161 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
8 ordsuc 7161 . . . 4 (Ord 𝐵 ↔ Ord suc 𝐵)
9 ordun 5972 . . . . 5 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10 ordelon 5890 . . . . . 6 ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
1110ex 397 . . . . 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 585 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14 ordssun 5970 . . . . . . 7 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
1514adantl 467 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵)))
16 ordsssuc 5955 . . . . . . 7 ((𝑥 ∈ On ∧ Ord (𝐴𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
171, 16sylan2 580 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴𝐵) ↔ 𝑥 ∈ suc (𝐴𝐵)))
18 ordsssuc 5955 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
1918adantrr 696 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐴𝑥 ∈ suc 𝐴))
20 ordsssuc 5955 . . . . . . . 8 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2120adantrl 695 . . . . . . 7 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥𝐵𝑥 ∈ suc 𝐵))
2219, 21orbi12d 904 . . . . . 6 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵)))
2315, 17, 223bitr3d 298 . . . . 5 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵)))
24 elun 3904 . . . . 5 (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ suc 𝐵))
2523, 24syl6bbr 278 . . . 4 ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
2625expcom 398 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
276, 13, 26pm5.21ndd 368 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
2827eqrdv 2769 1 ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  cun 3721  wss 3723  Ord word 5865  Oncon0 5866  suc csuc 5868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-suc 5872
This theorem is referenced by:  rankprb  8878  noetalem4  32203
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