Theorem oege2 44092

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 8585. (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 8473	. . . . 5 ⊢ 2o ∈ On
2		1oelpr 8470	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8467	. . . . . 6 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2864	. . . . 5 ⊢ 1o ∈ 2o
5		ondif2 8493	. . . . 5 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
6	1, 4, 5	mpbir2an 724	. . . 4 ⊢ 2o ∈ (On ∖ 2o)
7		oeworde 8585	. . . 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 8460	. . . 4 ⊢ 2o = suc 1o
11		onsucss 44051	. . . . . 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 3976	. . 3 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 2o ⊆ 𝐴)
15		simpll 779	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
16		onsseleq 6406	. . . . 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 8584	. . . . . 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 7426	. . . . . . 7 ⊢ (2o = 𝐴 → (2o ↑o 𝐵) = (𝐴 ↑o 𝐵))
21		ssid 3960	. . . . . . 7 ⊢ (𝐴 ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)
22	20, 21	eqsstrdi 3982	. . . . . 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 3948	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 2146   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  {cpr 4593  Oncon0 6364  suc csuc 6366  (class class class)co 7419  1_oc1o 8452  2_oc2o 8453   ↑_o coe 8458

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-omul 8464  df-oexp 8465

This theorem is used by: (None)