Theorem oaun3 43386

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 8578	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
2	1	difexd 5338	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) ∈ V)
3		uniprg 4929	. . . . 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 4484	. . . . 5 ⊢ (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 ∪ (𝐴 +o 𝐵))
6		oaword1 8595	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
7		ssequn1 4197	. . . . . 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 2788	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐴 +o 𝐵) ∖ 𝐴)) = (𝐴 +o 𝐵))
10	4, 9	eqtrd 2776	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, ((𝐴 +o 𝐵) ∖ 𝐴)} = (𝐴 +o 𝐵))
11		oaun3lem4 43381	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ∈ suc (𝐴 +o 𝐵))
12		unisng 4931	. . . 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 4180	. 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 4936	. . 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 4637	. . . . 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 43382	. . . . . . 7 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
18	17	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎})
19		oadif1 43384	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∖ 𝐴) = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +o 𝑏)})
20		eqidd 2737	. . . . . 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 4754	. . . . 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 2790	. . . 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 4926	. . 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 2790	. 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 43379	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
26		ssequn2 4200	. . 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 2785	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 1538   ∈ wcel 2107  {cab 2713  ∃wrex 3069  Vcvv 3479   ∖ cdif 3961   ∪ cun 3962   ⊆ wss 3964  {csn 4632  {cpr 4634  {ctp 4636  ∪ cuni 4913  Oncon0 6389  suc csuc 6391  (class class class)co 7435   +_o coa 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5286  ax-sep 5303  ax-nul 5313  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3435  df-v 3481  df-sbc 3793  df-csb 3910  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-pss 3984  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4914  df-int 4953  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5642  df-we 5644  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-ima 5703  df-pred 6326  df-ord 6392  df-on 6393  df-lim 6394  df-suc 6395  df-iota 6519  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7438  df-oprab 7439  df-mpo 7440  df-om 7892  df-2nd 8020  df-frecs 8311  df-wrecs 8342  df-recs 8416  df-rdg 8455  df-oadd 8515

This theorem is referenced by: (None)