Theorem oege2 43269

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 8534. (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 8424	. . . . 5 ⊢ 2o ∈ On
2		1oex 8421	. . . . . . 7 ⊢ 1o ∈ V
3	2	prid2 4723	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
4		df2o3 8419	. . . . . 6 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2827	. . . . 5 ⊢ 1o ∈ 2o
6		ondif2 8443	. . . . 5 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
7	1, 5, 6	mpbir2an 711	. . . 4 ⊢ 2o ∈ (On ∖ 2o)
8		oeworde 8534	. . . 4 ⊢ ((2o ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2o ↑o 𝐵))
9	7, 8	mpan 690	. . 3 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (2o ↑o 𝐵))
10	9	adantl 481	. 2 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2o ↑o 𝐵))
11		df-2o 8412	. . . 4 ⊢ 2o = suc 1o
12		onsucss 43228	. . . . . 6 ⊢ (𝐴 ∈ On → (1o ∈ 𝐴 → suc 1o ⊆ 𝐴))
13	12	imp 406	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1o ∈ 𝐴) → suc 1o ⊆ 𝐴)
14	13	adantr 480	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → suc 1o ⊆ 𝐴)
15	11, 14	eqsstrid 3982	. . 3 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 2o ⊆ 𝐴)
16		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
17		onsseleq 6361	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ⊆ 𝐴 ↔ (2o ∈ 𝐴 ∨ 2o = 𝐴)))
18	1, 16, 17	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 ↔ (2o ∈ 𝐴 ∨ 2o = 𝐴)))
19		oewordri 8533	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
20	19	adantlr 715	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ∈ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
21		oveq1 7376	. . . . . . 7 ⊢ (2o = 𝐴 → (2o ↑o 𝐵) = (𝐴 ↑o 𝐵))
22		ssid 3966	. . . . . . 7 ⊢ (𝐴 ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)
23	21, 22	eqsstrdi 3988	. . . . . 6 ⊢ (2o = 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵))
24	23	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o = 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
25	20, 24	jaod 859	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → ((2o ∈ 𝐴 ∨ 2o = 𝐴) → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
26	18, 25	sylbid 240	. . 3 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
27	15, 26	mpd 15	. 2 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵))
28	10, 27	sstrd 3954	1 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  {cpr 4587  Oncon0 6320  suc csuc 6322  (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: (None)