Theorem oege2 43738

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 8520. (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 8409	. . . . 5 ⊢ 2o ∈ On
2		1oex 8406	. . . . . . 7 ⊢ 1o ∈ V
3	2	prid2 4708	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
4		df2o3 8404	. . . . . 6 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2836	. . . . 5 ⊢ 1o ∈ 2o
6		ondif2 8428	. . . . 5 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
7	1, 5, 6	mpbir2an 712	. . . 4 ⊢ 2o ∈ (On ∖ 2o)
8		oeworde 8520	. . . 4 ⊢ ((2o ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2o ↑o 𝐵))
9	7, 8	mpan 691	. . 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 8397	. . . 4 ⊢ 2o = suc 1o
12		onsucss 43697	. . . . . 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 3961	. . 3 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 2o ⊆ 𝐴)
16		simpll 767	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
17		onsseleq 6356	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On) → (2o ⊆ 𝐴 ↔ (2o ∈ 𝐴 ∨ 2o = 𝐴)))
18	1, 16, 17	sylancr 588	. . . 4 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 ↔ (2o ∈ 𝐴 ∨ 2o = 𝐴)))
19		oewordri 8519	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
20	19	adantlr 716	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → (2o ∈ 𝐴 → (2o ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)))
21		oveq1 7365	. . . . . . 7 ⊢ (2o = 𝐴 → (2o ↑o 𝐵) = (𝐴 ↑o 𝐵))
22		ssid 3945	. . . . . . 7 ⊢ (𝐴 ↑o 𝐵) ⊆ (𝐴 ↑o 𝐵)
23	21, 22	eqsstrdi 3967	. . . . . 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 860	. . . 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 3933	1 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  {cpr 4570  Oncon0 6315  suc csuc 6317  (class class class)co 7358  1_oc1o 8389  2_oc2o 8390   ↑_o coe 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-omul 8401  df-oexp 8402

This theorem is referenced by: (None)