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 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))

Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 enrefg 8573 . . . . 5 (𝐴 ∈ On → 𝐴𝐴)
21adantr 484 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴𝐴)
3 simpr 488 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4 eqid 2759 . . . . . . . 8 (𝑥𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
54oacomf1olem 8207 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
65ancoms 462 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
76simpld 498 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
8 f1oeng 8560 . . . . 5 ((𝐵 ∈ On ∧ (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) → 𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
93, 7, 8syl2anc 587 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
10 incom 4109 . . . . 5 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
116simprd 499 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
1210, 11syl5eq 2806 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13 djuenun 9644 . . . 4 ((𝐴𝐴𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅) → (𝐴𝐵) ≈ (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
142, 9, 12, 13syl3anc 1369 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ≈ (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
15 oarec 8205 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
1614, 15breqtrrd 5065 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ≈ (𝐴 +o 𝐵))
1716ensymd 8592 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1539   ∈ wcel 2112   ∪ cun 3859   ∩ cin 3860  ∅c0 4228   class class class wbr 5037   ↦ cmpt 5117  ran crn 5530  Oncon0 6175  –1-1-onto→wf1o 6340  (class class class)co 7157   +o coa 8116   ≈ cen 8538   ⊔ cdju 9374 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-1o 8119  df-oadd 8123  df-er 8306  df-en 8542  df-dju 9377 This theorem is referenced by:  cardadju  9668  nnadjuALT  9672  tr3dom  40655
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