Theorem oeord2lim 43300

Description: Given a limit ordinal, the power of any base at least as large as two raised to an ordinal less than that limit ordinal is less than the power of that base raised to the limit ordinal . Lemma 3.22 of [Schloeder] p. 10. See oeordi 8604. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oeord2lim	⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))

Proof of Theorem oeord2lim

Step	Hyp	Ref	Expression
1		limelon 6422	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ Lim 𝐶) → 𝐶 ∈ On)
2	1	ancoms 458	. 2 ⊢ ((Lim 𝐶 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ On)
3		ondif2 8519	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
4	3	biimpri 228	. 2 ⊢ ((𝐴 ∈ On ∧ 1o ∈ 𝐴) → 𝐴 ∈ (On ∖ 2o))
5		oeordi 8604	. 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
6	2, 4, 5	syl2anr 597	1 ⊢ (((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ∖ cdif 3928  Oncon0 6357  Lim wlim 6358  (class class class)co 7410  1_oc1o 8478  2_oc2o 8479   ↑_o coe 8484

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490  df-oexp 8491

This theorem is referenced by: (None)