Theorem oaun3 44061

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 8523	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
2	1	difexd 5305	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V)
3		uniprg 4893	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)))
4	2, 3	syldan 602	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)))
5		undif2 4443	. . . . 5 ⊢ (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 ∪ (𝐴 +o 𝐵))
6		oaword1 8540	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
7		ssequn1 4147	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 +o 𝐵) ↔ (𝐴 ∪ (𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
8	6, 7	sylib 221	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ (𝐴 +o 𝐵)) = (𝐴 +o 𝐵))
9	5, 8	eqtrid 2817	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 +o 𝐵))
10	4, 9	eqtrd 2805	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 +o 𝐵))
11		oaun3lem4 44056	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵))
12		unisng 4895	. . . 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 18	. . 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 4131	. 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 4900	. . 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 4599	. . . . 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 44057	. . . . . . 7 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
18	17	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎})
19		oadif1 44059	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)})
20		eqidd 2771	. . . . . 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 4719	. . . . 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 2819	. . . 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 4890	. . 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 2819	. 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 44054	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
26		ssequn2 4150	. . 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 221	. 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 2814	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 400   = wceq 1568   ∈ wcel 2150  {cab 2748  ∃wrex 3096  Vcvv 3462   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  {csn 4594  {cpr 4596  {ctp 4598  ∪ cuni 4877  Oncon0 6364  suc csuc 6366  (class class class)co 7414   +_o coa 8453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-2nd 7990  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-oadd 8460

This theorem is referenced by: (None)