Theorem oewordi 8506

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 6316	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordgt0ge1 8408	. . . . . 6 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
4		1on 8397	. . . . . 6 ⊢ 1o ∈ On
5		onsseleq 6347	. . . . . 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 8417	. . . . . . 7 ⊢ (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o ∈ 𝐶))
10		oeword 8505	. . . . . . . . 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 8460	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = 1o)
18		oe1m 8460	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐵) = 1o)
20	17, 19	eqtr4d 2769	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = (1o ↑o 𝐵))
21		eqimss 3988	. . . . . . . 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 7353	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐴) = (𝐶 ↑o 𝐴))
24		oveq1 7353	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐵) = (𝐶 ↑o 𝐵))
25	23, 24	sseq12d 3963	. . . . . . 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 1541   ∈ wcel 2111   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4280  Ord word 6305  Oncon0 6306  (class class class)co 7346  1_oc1o 8378  2_oc2o 8379   ↑_o coe 8384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-omul 8390  df-oexp 8391

This theorem is referenced by:  oelim2  8510  oeoalem  8511  oeoelem  8513  oaabs2  8564  cantnflt  9562  cnfcom  9590  oege1  43409  cantnf2  43428  omabs2  43435  omltoe  43510