Theorem oewordi 8532

Description: Weak ordering property of ordinal exponentiation. Lemma 3.19 of [Schloeder] p. 10. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oewordi	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Proof of Theorem oewordi

Step	Hyp	Ref	Expression
1		eloni 6330	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordgt0ge1 8434	. . . . . 6 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
4		1on 8423	. . . . . 6 ⊢ 1o ∈ On
5		onsseleq 6361	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐶 ∈ On) → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
6	4, 5	mpan 690	. . . . 5 ⊢ (𝐶 ∈ On → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
7	3, 6	bitrd 279	. . . 4 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
8	7	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
9		ondif2 8443	. . . . . . 7 ⊢ (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o ∈ 𝐶))
10		oeword 8531	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
11	10	biimpd 229	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
12	11	3expia 1121	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
13	9, 12	biimtrrid 243	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1o ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
14	13	expd 415	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))))
15	14	3impia 1117	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
16		oe1m 8486	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = 1o)
18		oe1m 8486	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐵) = 1o)
20	17, 19	eqtr4d 2767	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = (1o ↑o 𝐵))
21		eqimss 4002	. . . . . . . 8 ⊢ ((1o ↑o 𝐴) = (1o ↑o 𝐵) → (1o ↑o 𝐴) ⊆ (1o ↑o 𝐵))
22	20, 21	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) ⊆ (1o ↑o 𝐵))
23		oveq1 7376	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐴) = (𝐶 ↑o 𝐴))
24		oveq1 7376	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐵) = (𝐶 ↑o 𝐵))
25	23, 24	sseq12d 3977	. . . . . . 7 ⊢ (1o = 𝐶 → ((1o ↑o 𝐴) ⊆ (1o ↑o 𝐵) ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
26	22, 25	syl5ibcom 245	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o = 𝐶 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
27	26	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o = 𝐶 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
28	27	a1dd 50	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o = 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
29	15, 28	jaod 859	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1o ∈ 𝐶 ∨ 1o = 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
30	8, 29	sylbid 240	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
31	30	imp 406	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  Ord word 6319  Oncon0 6320  (class class class)co 7369  1_oc1o 8404  2_oc2o 8405   ↑_o coe 8410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-omul 8416  df-oexp 8417

This theorem is referenced by:  oelim2  8536  oeoalem  8537  oeoelem  8539  oaabs2  8590  cantnflt  9601  cnfcom  9629  oege1  43288  cantnf2  43307  omabs2  43314  omltoe  43389