Theorem oawordri 8477

Description: Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)

Assertion

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

Proof of Theorem oawordri

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7366	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7366	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
3	1, 2	sseq12d 3966	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
4		oveq2 7366	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
5		oveq2 7366	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
6	4, 5	sseq12d 3966	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)))
7		oveq2 7366	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
8		oveq2 7366	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
9	7, 8	sseq12d 3966	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
10		oveq2 7366	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
11		oveq2 7366	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
12	10, 11	sseq12d 3966	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
13		oa0 8443	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
14	13	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o ∅) = 𝐴)
15		oa0 8443	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
16	15	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
17	14, 16	sseq12d 3966	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o ∅) ⊆ (𝐵 +o ∅) ↔ 𝐴 ⊆ 𝐵))
18	17	biimpar 477	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∅) ⊆ (𝐵 +o ∅))
19		oacl 8462	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o 𝑦) ∈ On)
20		eloni 6326	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝑦) ∈ On → Ord (𝐴 +o 𝑦))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐴 +o 𝑦))
22		oacl 8462	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
23		eloni 6326	. . . . . . . . . . 11 ⊢ ((𝐵 +o 𝑦) ∈ On → Ord (𝐵 +o 𝑦))
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐵 +o 𝑦))
25		ordsucsssuc 7765	. . . . . . . . . 10 ⊢ ((Ord (𝐴 +o 𝑦) ∧ Ord (𝐵 +o 𝑦)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
26	21, 24, 25	syl2an 597	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
27	26	anandirs 680	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
28		oasuc 8451	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
29	28	adantlr 716	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
30		oasuc 8451	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
31	30	adantll 715	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
32	29, 31	sseq12d 3966	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
33	27, 32	bitr4d 282	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
34	33	biimpd 229	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
35	34	expcom 413	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
36	35	adantrd 491	. . . 4 ⊢ (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
37		vex 3443	. . . . . . 7 ⊢ 𝑥 ∈ V
38		ss2iun 4964	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦))
39		oalim 8459	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
40	39	adantlr 716	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦))
41		oalim 8459	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦))
42	41	adantll 715	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦))
43	40, 42	sseq12d 3966	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐵 +o 𝑦)))
44	38, 43	imbitrrid 246	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)))
45	37, 44	mpanr1 704	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)))
46	45	expcom 413	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))))
47	46	adantrd 491	. . . 4 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (∀𝑦 ∈ 𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))))
48	3, 6, 9, 12, 18, 36, 47	tfinds3 7807	. . 3 ⊢ (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
49	48	exp4c 432	. 2 ⊢ (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))))
50	49	3imp231 1113	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3050  Vcvv 3439   ⊆ wss 3900  ∅c0 4284  ∪ ciun 4945  Ord word 6315  Oncon0 6316  Lim wlim 6317  suc csuc 6318  (class class class)co 7358   +_o coa 8394

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-oadd 8401

This theorem is referenced by:  oaword2  8480  omwordri  8499  oaabs2  8577  omabs2  43611  oaun3lem2  43654  naddwordnexlem4  43680