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Theorem onadju 10184
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 ∧ 𝐡 ∈ On) β†’ (𝐴 +o 𝐡) β‰ˆ (𝐴 βŠ” 𝐡))

Proof of Theorem onadju
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 enrefg 8976 . . . . 5 (𝐴 ∈ On β†’ 𝐴 β‰ˆ 𝐴)
21adantr 480 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐴 β‰ˆ 𝐴)
3 simpr 484 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐡 ∈ On)
4 eqid 2724 . . . . . . . 8 (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) = (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))
54oacomf1olem 8559 . . . . . . 7 ((𝐡 ∈ On ∧ 𝐴 ∈ On) β†’ ((π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∧ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…))
65ancoms 458 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∧ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…))
76simpld 494 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
8 f1oeng 8963 . . . . 5 ((𝐡 ∈ On ∧ (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) β†’ 𝐡 β‰ˆ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
93, 7, 8syl2anc 583 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐡 β‰ˆ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
10 incom 4193 . . . . 5 (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴)
116simprd 495 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…)
1210, 11eqtrid 2776 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = βˆ…)
13 djuenun 10161 . . . 4 ((𝐴 β‰ˆ 𝐴 ∧ 𝐡 β‰ˆ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∧ (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
142, 9, 12, 13syl3anc 1368 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
15 oarec 8557 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +o 𝐡) = (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
1614, 15breqtrrd 5166 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 +o 𝐡))
1716ensymd 8997 1 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +o 𝐡) β‰ˆ (𝐴 βŠ” 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3938   ∩ cin 3939  βˆ…c0 4314   class class class wbr 5138   ↦ cmpt 5221  ran crn 5667  Oncon0 6354  β€“1-1-ontoβ†’wf1o 6532  (class class class)co 7401   +o coa 8458   β‰ˆ cen 8932   βŠ” cdju 9889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-en 8936  df-dju 9892
This theorem is referenced by:  cardadju  10185  nnadjuALT  10189  tr3dom  42768
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