Theorem naddid1 33836

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 7282	. . 3 ⊢ (𝑎 = 𝑏 → (𝑎 +no ∅) = (𝑏 +no ∅))
2		id 22	. . 3 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
3	1, 2	eqeq12d 2754	. 2 ⊢ (𝑎 = 𝑏 → ((𝑎 +no ∅) = 𝑎 ↔ (𝑏 +no ∅) = 𝑏))
4		oveq1 7282	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no ∅) = (𝐴 +no ∅))
5		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	4, 5	eqeq12d 2754	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +no ∅) = 𝑎 ↔ (𝐴 +no ∅) = 𝐴))
7		0elon 6319	. . . . . 6 ⊢ ∅ ∈ On
8		naddov2 33834	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∅ ∈ On) → (𝑎 +no ∅) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)})
9	7, 8	mpan2 688	. . . . 5 ⊢ (𝑎 ∈ On → (𝑎 +no ∅) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)})
10	9	adantr 481	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (𝑎 +no ∅) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥)})
11		ral0 4443	. . . . . . . 8 ⊢ ∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥
12	11	biantrur 531	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ (∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥))
13		eleq1 2826	. . . . . . . . . . 11 ⊢ ((𝑏 +no ∅) = 𝑏 → ((𝑏 +no ∅) ∈ 𝑥 ↔ 𝑏 ∈ 𝑥))
14	13	ralimi 3087	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏 → ∀𝑏 ∈ 𝑎 ((𝑏 +no ∅) ∈ 𝑥 ↔ 𝑏 ∈ 𝑥))
15		ralbi 3089	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ 𝑥))
18		dfss3 3909	. . . . . . . 8 ⊢ (𝑎 ⊆ 𝑥 ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ 𝑥)
19	17, 18	bitr4di 289	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥 ↔ 𝑎 ⊆ 𝑥))
20	12, 19	bitr3id 285	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → ((∀𝑐 ∈ ∅ (𝑎 +no 𝑐) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) ∈ 𝑥) ↔ 𝑎 ⊆ 𝑥))
21	20	rabbidv 3414	. . . . 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 4884	. . . 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 4899	. . . . 5 ⊢ (𝑎 ∈ On → ∩ {𝑥 ∈ On ∣ 𝑎 ⊆ 𝑥} = 𝑎)
24	23	adantr 481	. . . 4 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → ∩ {𝑥 ∈ On ∣ 𝑎 ⊆ 𝑥} = 𝑎)
25	10, 22, 24	3eqtrd 2782	. . 3 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏) → (𝑎 +no ∅) = 𝑎)
26	25	ex 413	. 2 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 (𝑏 +no ∅) = 𝑏 → (𝑎 +no ∅) = 𝑎))
27	3, 6, 26	tfis3 7704	1 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ⊆ wss 3887  ∅c0 4256  ∩ cint 4879  Oncon0 6266  (class class class)co 7275   +no cnadd 33824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-nadd 33825

This theorem is referenced by: (None)