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Theorem onadju 10137
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 8930 . . . . 5 (𝐴 ∈ On β†’ 𝐴 β‰ˆ 𝐴)
21adantr 482 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐴 β‰ˆ 𝐴)
3 simpr 486 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐡 ∈ On)
4 eqid 2733 . . . . . . . 8 (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) = (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))
54oacomf1olem 8515 . . . . . . 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 460 . . . . . 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 496 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)):𝐡–1-1-ontoβ†’ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
8 f1oeng 8917 . . . . 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 585 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐡 β‰ˆ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)))
10 incom 4165 . . . . 5 (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴)
116simprd 497 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯)) ∩ 𝐴) = βˆ…)
1210, 11eqtrid 2785 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 ∩ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))) = βˆ…)
13 djuenun 10114 . . . 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 1372 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
15 oarec 8513 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 +o 𝐡) = (𝐴 βˆͺ ran (π‘₯ ∈ 𝐡 ↦ (𝐴 +o π‘₯))))
1614, 15breqtrrd 5137 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 +o 𝐡))
1716ensymd 8951 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 397   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3912   ∩ cin 3913  βˆ…c0 4286   class class class wbr 5109   ↦ cmpt 5192  ran crn 5638  Oncon0 6321  β€“1-1-ontoβ†’wf1o 6499  (class class class)co 7361   +o coa 8413   β‰ˆ cen 8886   βŠ” cdju 9842
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420  df-er 8654  df-en 8890  df-dju 9845
This theorem is referenced by:  cardadju  10138  nnadjuALT  10142  tr3dom  41892
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