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Theorem onadju 10175
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 8968 . . . . 5 (𝐴 ∈ On → 𝐴𝐴)
21adantr 482 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴𝐴)
3 simpr 486 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4 eqid 2733 . . . . . . . 8 (𝑥𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
54oacomf1olem 8552 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
65ancoms 460 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
76simpld 496 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
8 f1oeng 8955 . . . . 5 ((𝐵 ∈ On ∧ (𝑥𝐵 ↦ (𝐴 +o 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) → 𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
93, 7, 8syl2anc 585 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
10 incom 4199 . . . . 5 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
116simprd 497 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
1210, 11eqtrid 2785 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13 djuenun 10152 . . . 4 ((𝐴𝐴𝐵 ≈ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)) ∧ (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))) = ∅) → (𝐴𝐵) ≈ (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
142, 9, 12, 13syl3anc 1372 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ≈ (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
15 oarec 8550 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
1614, 15breqtrrd 5172 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ≈ (𝐴 +o 𝐵))
1716ensymd 8989 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cun 3944  cin 3945  c0 4320   class class class wbr 5144  cmpt 5227  ran crn 5673  Oncon0 6356  1-1-ontowf1o 6534  (class class class)co 7396   +o coa 8450  cen 8924  cdju 9880
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8691  df-en 8928  df-dju 9883
This theorem is referenced by:  cardadju  10176  nnadjuALT  10180  tr3dom  42150
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