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Theorem oa0r 7570
Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
Assertion
Ref Expression
oa0r (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)

Proof of Theorem oa0r
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6618 . . 3 (𝑥 = ∅ → (∅ +𝑜 𝑥) = (∅ +𝑜 ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2636 . 2 (𝑥 = ∅ → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 ∅) = ∅))
4 oveq2 6618 . . 3 (𝑥 = 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2636 . 2 (𝑥 = 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝑦) = 𝑦))
7 oveq2 6618 . . 3 (𝑥 = suc 𝑦 → (∅ +𝑜 𝑥) = (∅ +𝑜 suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2636 . 2 (𝑥 = suc 𝑦 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 suc 𝑦) = suc 𝑦))
10 oveq2 6618 . . 3 (𝑥 = 𝐴 → (∅ +𝑜 𝑥) = (∅ +𝑜 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2636 . 2 (𝑥 = 𝐴 → ((∅ +𝑜 𝑥) = 𝑥 ↔ (∅ +𝑜 𝐴) = 𝐴))
13 0elon 5742 . . 3 ∅ ∈ On
14 oa0 7548 . . 3 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +𝑜 ∅) = ∅
16 oasuc 7556 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
1713, 16mpan 705 . . . 4 (𝑦 ∈ On → (∅ +𝑜 suc 𝑦) = suc (∅ +𝑜 𝑦))
18 suceq 5754 . . . 4 ((∅ +𝑜 𝑦) = 𝑦 → suc (∅ +𝑜 𝑦) = suc 𝑦)
1917, 18sylan9eq 2675 . . 3 ((𝑦 ∈ On ∧ (∅ +𝑜 𝑦) = 𝑦) → (∅ +𝑜 suc 𝑦) = suc 𝑦)
2019ex 450 . 2 (𝑦 ∈ On → ((∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 suc 𝑦) = suc 𝑦))
21 iuneq2 4508 . . . 4 (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 4544 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22syl6eqr 2673 . . 3 (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑥)
24 vex 3192 . . . . 5 𝑥 ∈ V
25 oalim 7564 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
2613, 25mpan 705 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
2724, 26mpan 705 . . . 4 (Lim 𝑥 → (∅ +𝑜 𝑥) = 𝑦𝑥 (∅ +𝑜 𝑦))
28 limuni 5749 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2636 . . 3 (Lim 𝑥 → ((∅ +𝑜 𝑥) = 𝑥 𝑦𝑥 (∅ +𝑜 𝑦) = 𝑥))
3023, 29syl5ibr 236 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +𝑜 𝑦) = 𝑦 → (∅ +𝑜 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 7013 1 (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  c0 3896   cuni 4407   ciun 4490  Oncon0 5687  Lim wlim 5688  suc csuc 5689  (class class class)co 6610   +𝑜 coa 7509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516
This theorem is referenced by:  om1  7574  oaword2  7585  oeeui  7634  oaabs2  7677  cantnfp1  8530
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