Theorem oawordi 5924

Description: Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)

Assertion

Ref	Expression
oawordi	⊢ ((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) → (A ⊆ B → (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))

Proof of Theorem oawordi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oafnex 5904	. . . . 5 ⊢ (x ∈ V ↦ suc x) Fn V
2	1	a1i 9	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (x ∈ V ↦ suc x) Fn V)
3		simpl3 899	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → 𝐶 ∈ On)
4		simpl1 897	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → A ∈ On)
5		simpl2 898	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → B ∈ On)
6		ax-ia2 100	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → A ⊆ B)
7	2, 3, 4, 5, 6	rdgss 5855	. . 3 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (rec((x ∈ V ↦ suc x), 𝐶)‘A) ⊆ (rec((x ∈ V ↦ suc x), 𝐶)‘B))
8	3, 4	jca 290	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 ∈ On ∧ A ∈ On))
9		oav 5913	. . . 4 ⊢ ((𝐶 ∈ On ∧ A ∈ On) → (𝐶 +𝑜 A) = (rec((x ∈ V ↦ suc x), 𝐶)‘A))
10	8, 9	syl 14	. . 3 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 +𝑜 A) = (rec((x ∈ V ↦ suc x), 𝐶)‘A))
11	3, 5	jca 290	. . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 ∈ On ∧ B ∈ On))
12		oav 5913	. . . 4 ⊢ ((𝐶 ∈ On ∧ B ∈ On) → (𝐶 +𝑜 B) = (rec((x ∈ V ↦ suc x), 𝐶)‘B))
13	11, 12	syl 14	. . 3 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 +𝑜 B) = (rec((x ∈ V ↦ suc x), 𝐶)‘B))
14	7, 10, 13	3sstr4d 2966	. 2 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B))
15	14	ex 108	1 ⊢ ((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) → (A ⊆ B → (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 875   = wceq 1373   ∈ wcel 1375  Vcvv 2533   ⊆ wss 2895   ∈ cmpt 3770  Oncon0 4024  suc csuc 4026   Fn wfn 4791  ‘cfv 4796  (class class class)co 5405  reccrdg 5842   +_𝑜 coa 5878

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-coll 3824  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093  ax-setind 4174

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-fal 1232  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-csb 2829  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-tr 3807  df-id 3983  df-iord 4027  df-on 4028  df-suc 4031  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-f1 4801  df-fo 4802  df-f1o 4803  df-fv 4804  df-ov 5408  df-oprab 5409  df-mpt2 5410  df-recs 5807  df-irdg 5843  df-oadd 5885

This theorem is referenced by: (None)