Theorem oaun3 43566

Description: Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)

Assertion

Ref	Expression
oaun3	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})

Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧

Proof of Theorem oaun3

Step	Hyp	Ref	Expression
1		oacl 8460	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
2	1	difexd 5274	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V)
3		uniprg 4877	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)))
4	2, 3	syldan 591	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)))
5		undif2 4427	. . . . 5 ⊢ (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 ∪ (𝐴 +o 𝐵))
6		oaword1 8477	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
7		ssequn1 4136	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 +o 𝐵) ↔ (𝐴 ∪ (𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
8	6, 7	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ (𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
9	5, 8	eqtrid 2781	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 +o 𝐵))
10	4, 9	eqtrd 2769	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 +o 𝐵))
11		oaun3lem4 43561	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵))
12		unisng 4879	. . . 4 ⊢ ({𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵) → ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)})
13	11, 12	syl 17	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)})
14	10, 13	uneq12d 4119	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = ((𝐴 +o 𝐵) ∪ {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}))
15		uniun 4884	. . 3 ⊢ ∪ ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = (∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
16		df-tp 4583	. . . . 5 ⊢ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴), {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
17		rp-abid 43562	. . . . . . 7 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
18	17	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎})
19		oadif1 43564	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)})
20		eqidd 2735	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)})
21	18, 19, 20	tpeq123d 4703	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴), {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}} = {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
22	16, 21	eqtr3id 2783	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
23	22	unieqd 4874	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ ({𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
24	15, 23	eqtr3id 2783	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} ∪ ∪ {{𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}}) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})
25		oaun3lem2 43559	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
26		ssequn2 4139	. . 3 ⊢ ({𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵) ↔ ((𝐴 +o 𝐵) ∪ {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}) = (𝐴 +o 𝐵))
27	25, 26	sylib 218	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∪ {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}) = (𝐴 +o 𝐵))
28	14, 24, 27	3eqtr3rd 2778	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)}})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  ∃wrex 3058  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {csn 4578  {cpr 4580  {ctp 4582  ∪ cuni 4861  Oncon0 6315  suc csuc 6317  (class class class)co 7356   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399

This theorem is referenced by: (None)