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Theorem oa0r 8553
Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. Lemma 2.14 of [Schloeder] p. 5. (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 7423 . . 3 (𝑥 = ∅ → (∅ +o 𝑥) = (∅ +o ∅))
2 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
31, 2eqeq12d 2744 . 2 (𝑥 = ∅ → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o ∅) = ∅))
4 oveq2 7423 . . 3 (𝑥 = 𝑦 → (∅ +o 𝑥) = (∅ +o 𝑦))
5 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
64, 5eqeq12d 2744 . 2 (𝑥 = 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝑦) = 𝑦))
7 oveq2 7423 . . 3 (𝑥 = suc 𝑦 → (∅ +o 𝑥) = (∅ +o suc 𝑦))
8 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
97, 8eqeq12d 2744 . 2 (𝑥 = suc 𝑦 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o suc 𝑦) = suc 𝑦))
10 oveq2 7423 . . 3 (𝑥 = 𝐴 → (∅ +o 𝑥) = (∅ +o 𝐴))
11 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2744 . 2 (𝑥 = 𝐴 → ((∅ +o 𝑥) = 𝑥 ↔ (∅ +o 𝐴) = 𝐴))
13 0elon 6418 . . 3 ∅ ∈ On
14 oa0 8531 . . 3 (∅ ∈ On → (∅ +o ∅) = ∅)
1513, 14ax-mp 5 . 2 (∅ +o ∅) = ∅
16 oasuc 8539 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
1713, 16mpan 689 . . . 4 (𝑦 ∈ On → (∅ +o suc 𝑦) = suc (∅ +o 𝑦))
18 suceq 6430 . . . 4 ((∅ +o 𝑦) = 𝑦 → suc (∅ +o 𝑦) = suc 𝑦)
1917, 18sylan9eq 2788 . . 3 ((𝑦 ∈ On ∧ (∅ +o 𝑦) = 𝑦) → (∅ +o suc 𝑦) = suc 𝑦)
2019ex 412 . 2 (𝑦 ∈ On → ((∅ +o 𝑦) = 𝑦 → (∅ +o suc 𝑦) = suc 𝑦))
21 iuneq2 5011 . . . 4 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑦𝑥 𝑦)
22 uniiun 5056 . . . 4 𝑥 = 𝑦𝑥 𝑦
2321, 22eqtr4di 2786 . . 3 (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 𝑦𝑥 (∅ +o 𝑦) = 𝑥)
24 vex 3474 . . . . 5 𝑥 ∈ V
25 oalim 8547 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2613, 25mpan 689 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
2724, 26mpan 689 . . . 4 (Lim 𝑥 → (∅ +o 𝑥) = 𝑦𝑥 (∅ +o 𝑦))
28 limuni 6425 . . . 4 (Lim 𝑥𝑥 = 𝑥)
2927, 28eqeq12d 2744 . . 3 (Lim 𝑥 → ((∅ +o 𝑥) = 𝑥 𝑦𝑥 (∅ +o 𝑦) = 𝑥))
3023, 29imbitrrid 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ +o 𝑦) = 𝑦 → (∅ +o 𝑥) = 𝑥))
313, 6, 9, 12, 15, 20, 30tfinds 7859 1 (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3057  Vcvv 3470  c0 4319   cuni 4904   ciun 4992  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7415   +o coa 8478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-2nd 7989  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-oadd 8485
This theorem is referenced by:  om1  8557  oaword2  8568  oeeui  8617  oaabs2  8664  cantnfp1  9699  oaordnrex  42715  oacl2g  42750  tfsconcat0i  42765  ofoaf  42775  ofoaid2  42779
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