Theorem naddid1 33433

Description: Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)

Assertion

Ref	Expression
naddid1	⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)

Proof of Theorem naddid1

Dummy variables 𝑎 𝑏 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7163	. . 3 ⊢ (𝑎 = 𝑏 → (𝑎 +no ∅) = (𝑏 +no ∅))
2		id 22	. . 3 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
3	1, 2	eqeq12d 2774	. 2 ⊢ (𝑎 = 𝑏 → ((𝑎 +no ∅) = 𝑎 ↔ (𝑏 +no ∅) = 𝑏))
4		oveq1 7163	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no ∅) = (𝐴 +no ∅))
5		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	4, 5	eqeq12d 2774	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +no ∅) = 𝑎 ↔ (𝐴 +no ∅) = 𝐴))
7		0elon 6227	. . . . . 6 ⊢ ∅ ∈ On
8		naddov2 33431	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∅ ∈ On) → (𝑎 +no ∅) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)})
9	7, 8	mpan2 690	. . . . 5 ⊢ (𝑎 ∈ On → (𝑎 +no ∅) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)})
10	9	adantr 484	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (𝑎 +no ∅) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)})
11		ral0 4408	. . . . . . . 8 ⊢ ∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥
12	11	biantrur 534	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥))
13		eleq1 2839	. . . . . . . . . . 11 ⊢ ((𝑏 +no ∅) = 𝑏 → ((𝑏 +no ∅) ∈ 𝑥 ↔ 𝑏 ∈ 𝑥))
14	13	ralimi 3092	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏 → ∀𝑏 ∈ 𝑎 ((𝑏 +no ∅) ∈ 𝑥 ↔ 𝑏 ∈ 𝑥))
15		ralbi 3099	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑎 ((𝑏 +no ∅) ∈ 𝑥 ↔ 𝑏 ∈ 𝑥) → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ 𝑥))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏 → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ 𝑥))
17	16	adantl 485	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ 𝑥))
18		dfss3 3882	. . . . . . . 8 ⊢ (𝑎 ⊆ 𝑥 ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ 𝑥)
19	17, 18	bitr4di 292	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ 𝑎 ⊆ 𝑥))
20	12, 19	bitr3id 288	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → ((∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥) ↔ 𝑎 ⊆ 𝑥))
21	20	rabbidv 3392	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)} = {𝑥 ∈ On ∣ 𝑎 ⊆ 𝑥})
22	21	inteqd 4846	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)} = ∩ {𝑥 ∈ On ∣ 𝑎 ⊆ 𝑥})
23		intmin 4861	. . . . 5 ⊢ (𝑎 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑎 ⊆ 𝑥} = 𝑎)
24	23	adantr 484	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → ∩ {𝑥 ∈ On ∣ 𝑎 ⊆ 𝑥} = 𝑎)
25	10, 22, 24	3eqtrd 2797	. . 3 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (𝑎 +no ∅) = 𝑎)
26	25	ex 416	. 2 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏 → (𝑎 +no ∅) = 𝑎))
27	3, 6, 26	tfis3 7577	1 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   ⊆ wss 3860  ∅c0 4227  ∩ cint 4841  Oncon0 6174  (class class class)co 7156   +no cnadd 33421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-frecs 33392  df-nadd 33422

This theorem is referenced by: (None)