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Theorem onadju 10187
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 8979 . . . . 5 (𝐴 ∈ On β†’ 𝐴 β‰ˆ 𝐴)
21adantr 481 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐴 β‰ˆ 𝐴)
3 simpr 485 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐡 ∈ On)
4 eqid 2732 . . . . . . . 8 (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) = (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))
54oacomf1olem 8563 . . . . . . 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 459 . . . . . 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 495 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
8 f1oeng 8966 . . . . 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 584 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐡 β‰ˆ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
10 incom 4201 . . . . 5 (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴)
116simprd 496 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…)
1210, 11eqtrid 2784 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = βˆ…)
13 djuenun 10164 . . . 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 1371 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
15 oarec 8561 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +o 𝐡) = (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
1614, 15breqtrrd 5176 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 +o 𝐡))
1716ensymd 9000 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 396   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3946   ∩ cin 3947  βˆ…c0 4322   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  Oncon0 6364  β€“1-1-ontoβ†’wf1o 6542  (class class class)co 7408   +o coa 8462   β‰ˆ cen 8935   βŠ” cdju 9892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-oadd 8469  df-er 8702  df-en 8939  df-dju 9895
This theorem is referenced by:  cardadju  10188  nnadjuALT  10192  tr3dom  42269
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