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Theorem onadju 10232
Description: The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Jim Kingdon, 7-Sep-2023.)
Assertion
Ref Expression
onadju ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴𝐵))

Proof of Theorem onadju
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 enrefg 9023 . . . . 5 (𝐴 ∈ On → 𝐴𝐴)
21adantr 480 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴𝐴)
3 simpr 484 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4 eqid 2735 . . . . . . . 8 (𝑥𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
54oacomf1olem 8601 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
65ancoms 458 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
76simpld 494 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
8 f1oeng 9010 . . . . 5 ((𝐵 ∈ On ∧ (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) → 𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
93, 7, 8syl2anc 584 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
10 incom 4217 . . . . 5 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
116simprd 495 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
1210, 11eqtrid 2787 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13 djuenun 10209 . . . 4 ((𝐴𝐴𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅) → (𝐴𝐵) ≈ (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
142, 9, 12, 13syl3anc 1370 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ≈ (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
15 oarec 8599 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
1614, 15breqtrrd 5176 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ≈ (𝐴 +o 𝐵))
1716ensymd 9044 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  cin 3962  c0 4339   class class class wbr 5148  cmpt 5231  ran crn 5690  Oncon0 6386  1-1-ontowf1o 6562  (class class class)co 7431   +o coa 8502  cen 8981  cdju 9936
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dju 9939
This theorem is referenced by:  cardadju  10233  nnadjuALT  10237  tr3dom  43518
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