Theorem oege2 44149

Description: Any power of an ordinal at least as large as two is greater-than-or-equal to the term on the right. Lemma 3.20 of [Schloeder] p. 10. See oeworde 8582. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
oege2	⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Proof of Theorem oege2

Step	Hyp	Ref	Expression
1		2on 8470	. . . . 5 ⊢ 2o ∈ On
2		1oelpr 8467	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8464	. . . . . 6 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2859	. . . . 5 ⊢ 1o ∈ 2o
5		ondif2 8490	. . . . 5 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
6	1, 4, 5	mpbir2an 724	. . . 4 ⊢ 2o ∈ (On ∖ 2o)
7		oeworde 8582	. . . 4 ⊢ ((2o ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2o ↑o 𝐵))
8	6, 7	mpan 703	. . 3 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (2o ↑o 𝐵))
9	8	adantl 487	. 2 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2o ↑o 𝐵))
10		df-2o 8457	. . . 4 ⊢ 2o = suc 1o
11		onsucss 44108	. . . . . 6 ⊢ (𝐴 ∈ On → (1o ∈ 𝐴 → suc 1o ⊆ 𝐴))
12	11	imp 412	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1o ∈ 𝐴) → suc 1o ⊆ 𝐴)
13	12	adantr 486	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → suc 1o ⊆ 𝐴)
14	10, 13	eqsstrid 3969	. . 3 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 2o ⊆ 𝐴)
15		simpll 779	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
16		onsseleq 6399	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ⊆ 𝐴 ↔ (2o ∈ 𝐴 ∨ 2o = 𝐴)))
17	1, 15, 16	sylancr 599	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 ↔ (2o ∈ 𝐴 ∨ 2o = 𝐴)))
18		oewordri 8581	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
19	18	adantlr 728	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ∈ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
20		oveq1 7421	. . . . . . 7 ⊢ (2o = 𝐴 → (2o ↑o 𝐵) = (𝐴 ↑o 𝐵))
21		ssid 3953	. . . . . . 7 ⊢ (𝐴 ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)
22	20, 21	eqsstrdi 3975	. . . . . 6 ⊢ (2o = 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵))
23	22	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o = 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
24	19, 23	jaod 873	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → ((2o ∈ 𝐴 ∨ 2o = 𝐴) → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
25	17, 24	sylbid 243	. . 3 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
26	14, 25	mpd 16	. 2 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵))
27	9, 26	sstrd 3941	1 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  {cpr 4586  Oncon0 6357  suc csuc 6359  (class class class)co 7414  1_oc1o 8449  2_oc2o 8450   ↑_o coe 8455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-oadd 8460  df-omul 8461  df-oexp 8462

This theorem is used by: (None)