Theorem oewordi 8562

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 6357	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordgt0ge1 8463	. . . . . 6 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
4		1on 8451	. . . . . 6 ⊢ 1o ∈ On
5		onsseleq 6388	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐶 ∈ On) → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
6	4, 5	mpan 700	. . . . 5 ⊢ (𝐶 ∈ On → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
7	3, 6	bitrd 281	. . . 4 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
8	7	3ad2ant3 1149	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
9		ondif2 8472	. . . . . . 7 ⊢ (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o ∈ 𝐶))
10		oeword 8561	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
11	10	biimpd 231	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
12	11	3expia 1135	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
13	9, 12	biimtrrid 245	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1o ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
14	13	expd 419	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))))
15	14	3impia 1131	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
16		oe1m 8515	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
17	16	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = 1o)
18		oe1m 8515	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
19	18	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐵) = 1o)
20	17, 19	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = (1o ↑o 𝐵))
21		eqimss 3995	. . . . . . . 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 7404	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐴) = (𝐶 ↑o 𝐴))
24		oveq1 7404	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐵) = (𝐶 ↑o 𝐵))
25	23, 24	sseq12d 3970	. . . . . . 7 ⊢ (1o = 𝐶 → ((1o ↑o 𝐴) ⊆ (1o ↑o 𝐵) ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
26	22, 25	syl5ibcom 247	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o = 𝐶 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
27	26	3adant3 1146	. . . . 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 870	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1o ∈ 𝐶 ∨ 1o = 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
30	8, 29	sylbid 242	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
31	30	imp 410	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  Ord word 6346  Oncon0 6347  (class class class)co 7397  1_oc1o 8431  2_oc2o 8432   ↑_o coe 8437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-oadd 8442  df-omul 8443  df-oexp 8444

This theorem is referenced by:  oelim2  8566  oeoalem  8567  oeoelem  8569  oaabs2  8620  cantnflt  9628  cnfcom  9656  oege1  43884  cantnf2  43903  omabs2  43910  omltoe  43984