Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnadju Structured version   Visualization version   GIF version

 Description: The cardinal and ordinal sums of finite ordinals are equal. For a shorter proof using ax-rep 5160, see nnadjuALT 9671. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.) Avoid ax-rep 5160. (Revised by BTernaryTau, 2-Jul-2024.)
Assertion
Ref Expression
nnadju ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴𝐵)) = (𝐴 +o 𝐵))

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djueq2 9381 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑥) = (𝐴𝐵))
2 oveq2 7164 . . . . . . 7 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
31, 2breq12d 5049 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴𝐵) ≈ (𝐴 +o 𝐵)))
43imbi2d 344 . . . . 5 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑥) ≈ (𝐴 +o 𝑥)) ↔ (𝐴 ∈ ω → (𝐴𝐵) ≈ (𝐴 +o 𝐵))))
5 djueq2 9381 . . . . . . 7 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ⊔ ∅))
6 oveq2 7164 . . . . . . 7 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
75, 6breq12d 5049 . . . . . 6 (𝑥 = ∅ → ((𝐴𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴 ⊔ ∅) ≈ (𝐴 +o ∅)))
8 djueq2 9381 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
9 oveq2 7164 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
108, 9breq12d 5049 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴𝑦) ≈ (𝐴 +o 𝑦)))
11 djueq2 9381 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴𝑥) = (𝐴 ⊔ suc 𝑦))
12 oveq2 7164 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
1311, 12breq12d 5049 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦)))
14 dju0en 9648 . . . . . . 7 (𝐴 ∈ ω → (𝐴 ⊔ ∅) ≈ 𝐴)
15 nna0 8246 . . . . . . 7 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
1614, 15breqtrrd 5064 . . . . . 6 (𝐴 ∈ ω → (𝐴 ⊔ ∅) ≈ (𝐴 +o ∅))
17 1oex 8126 . . . . . . . . . . 11 1o ∈ V
18 djuassen 9651 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω ∧ 1o ∈ V) → ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 ⊔ (𝑦 ⊔ 1o)))
1917, 18mp3an3 1447 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 ⊔ (𝑦 ⊔ 1o)))
20 enrefg 8572 . . . . . . . . . . 11 (𝐴 ∈ ω → 𝐴𝐴)
21 nnord 7593 . . . . . . . . . . . . 13 (𝑦 ∈ ω → Ord 𝑦)
22 ordirr 6192 . . . . . . . . . . . . 13 (Ord 𝑦 → ¬ 𝑦𝑦)
2321, 22syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ω → ¬ 𝑦𝑦)
24 dju1en 9644 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ¬ 𝑦𝑦) → (𝑦 ⊔ 1o) ≈ suc 𝑦)
2523, 24mpdan 686 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 ⊔ 1o) ≈ suc 𝑦)
26 djuen 9642 . . . . . . . . . . 11 ((𝐴𝐴 ∧ (𝑦 ⊔ 1o) ≈ suc 𝑦) → (𝐴 ⊔ (𝑦 ⊔ 1o)) ≈ (𝐴 ⊔ suc 𝑦))
2720, 25, 26syl2an 598 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊔ (𝑦 ⊔ 1o)) ≈ (𝐴 ⊔ suc 𝑦))
28 entr 8592 . . . . . . . . . 10 ((((𝐴𝑦) ⊔ 1o) ≈ (𝐴 ⊔ (𝑦 ⊔ 1o)) ∧ (𝐴 ⊔ (𝑦 ⊔ 1o)) ≈ (𝐴 ⊔ suc 𝑦)) → ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 ⊔ suc 𝑦))
2919, 27, 28syl2anc 587 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 ⊔ suc 𝑦))
3029ensymd 8591 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊔ suc 𝑦) ≈ ((𝐴𝑦) ⊔ 1o))
3117enref 8573 . . . . . . . . . . . 12 1o ≈ 1o
32 djuen 9642 . . . . . . . . . . . 12 (((𝐴𝑦) ≈ (𝐴 +o 𝑦) ∧ 1o ≈ 1o) → ((𝐴𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o))
3331, 32mpan2 690 . . . . . . . . . . 11 ((𝐴𝑦) ≈ (𝐴 +o 𝑦) → ((𝐴𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o))
3433a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑦) ≈ (𝐴 +o 𝑦) → ((𝐴𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o)))
35 nnacl 8253 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
36 nnord 7593 . . . . . . . . . . . . 13 ((𝐴 +o 𝑦) ∈ ω → Ord (𝐴 +o 𝑦))
37 ordirr 6192 . . . . . . . . . . . . 13 (Ord (𝐴 +o 𝑦) → ¬ (𝐴 +o 𝑦) ∈ (𝐴 +o 𝑦))
3835, 36, 373syl 18 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ¬ (𝐴 +o 𝑦) ∈ (𝐴 +o 𝑦))
39 dju1en 9644 . . . . . . . . . . . 12 (((𝐴 +o 𝑦) ∈ ω ∧ ¬ (𝐴 +o 𝑦) ∈ (𝐴 +o 𝑦)) → ((𝐴 +o 𝑦) ⊔ 1o) ≈ suc (𝐴 +o 𝑦))
4035, 38, 39syl2anc 587 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑦) ⊔ 1o) ≈ suc (𝐴 +o 𝑦))
41 nnasuc 8248 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
4240, 41breqtrrd 5064 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦))
4334, 42jctird 530 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑦) ≈ (𝐴 +o 𝑦) → (((𝐴𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o) ∧ ((𝐴 +o 𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦))))
44 entr 8592 . . . . . . . . 9 ((((𝐴𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o) ∧ ((𝐴 +o 𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦)) → ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦))
4543, 44syl6 35 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑦) ≈ (𝐴 +o 𝑦) → ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦)))
46 entr 8592 . . . . . . . 8 (((𝐴 ⊔ suc 𝑦) ≈ ((𝐴𝑦) ⊔ 1o) ∧ ((𝐴𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦)) → (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦))
4730, 45, 46syl6an 683 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑦) ≈ (𝐴 +o 𝑦) → (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦)))
4847expcom 417 . . . . . 6 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑦) ≈ (𝐴 +o 𝑦) → (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦))))
497, 10, 13, 16, 48finds2 7616 . . . . 5 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑥) ≈ (𝐴 +o 𝑥)))
504, 49vtoclga 3494 . . . 4 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵) ≈ (𝐴 +o 𝐵)))
5150impcom 411 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵) ≈ (𝐴 +o 𝐵))
52 carden2b 9442 . . 3 ((𝐴𝐵) ≈ (𝐴 +o 𝐵) → (card‘(𝐴𝐵)) = (card‘(𝐴 +o 𝐵)))
5351, 52syl 17 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴𝐵)) = (card‘(𝐴 +o 𝐵)))
54 nnacl 8253 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
55 cardnn 9438 . . 3 ((𝐴 +o 𝐵) ∈ ω → (card‘(𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
5654, 55syl 17 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
5753, 56eqtrd 2793 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴𝐵)) = (𝐴 +o 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227   class class class wbr 5036  Ord word 6173  suc csuc 6176  ‘cfv 6340  (class class class)co 7156  ωcom 7585  1oc1o 8111   +o coa 8115   ≈ cen 8537   ⊔ cdju 9373  cardccrd 9410 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-oadd 8122  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-dju 9376  df-card 9414 This theorem is referenced by:  ficardadju  9672  ackbij1lem5  9697  ackbij1lem9  9701
 Copyright terms: Public domain W3C validator