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Theorem oege2 43296
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 8629. (Contributed by RP, 29-Jan-2025.)
Assertion
Ref Expression
oege2 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))

Proof of Theorem oege2
StepHypRef Expression
1 2on 8518 . . . . 5 2o ∈ On
2 1oex 8514 . . . . . . 7 1o ∈ V
32prid2 4767 . . . . . 6 1o ∈ {∅, 1o}
4 df2o3 8512 . . . . . 6 2o = {∅, 1o}
53, 4eleqtrri 2837 . . . . 5 1o ∈ 2o
6 ondif2 8538 . . . . 5 (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
71, 5, 6mpbir2an 711 . . . 4 2o ∈ (On ∖ 2o)
8 oeworde 8629 . . . 4 ((2o ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2oo 𝐵))
97, 8mpan 690 . . 3 (𝐵 ∈ On → 𝐵 ⊆ (2oo 𝐵))
109adantl 481 . 2 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (2oo 𝐵))
11 df-2o 8505 . . . 4 2o = suc 1o
12 onsucss 43255 . . . . . 6 (𝐴 ∈ On → (1o𝐴 → suc 1o𝐴))
1312imp 406 . . . . 5 ((𝐴 ∈ On ∧ 1o𝐴) → suc 1o𝐴)
1413adantr 480 . . . 4 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → suc 1o𝐴)
1511, 14eqsstrid 4043 . . 3 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 2o𝐴)
16 simpll 767 . . . . 5 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
17 onsseleq 6426 . . . . 5 ((2o ∈ On ∧ 𝐴 ∈ On) → (2o𝐴 ↔ (2o𝐴 ∨ 2o = 𝐴)))
181, 16, 17sylancr 587 . . . 4 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → (2o𝐴 ↔ (2o𝐴 ∨ 2o = 𝐴)))
19 oewordri 8628 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵)))
2019adantlr 715 . . . . 5 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → (2o𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵)))
21 oveq1 7437 . . . . . . 7 (2o = 𝐴 → (2oo 𝐵) = (𝐴o 𝐵))
22 ssid 4017 . . . . . . 7 (𝐴o 𝐵) ⊆ (𝐴o 𝐵)
2321, 22eqsstrdi 4049 . . . . . 6 (2o = 𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵))
2423a1i 11 . . . . 5 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → (2o = 𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵)))
2520, 24jaod 859 . . . 4 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → ((2o𝐴 ∨ 2o = 𝐴) → (2oo 𝐵) ⊆ (𝐴o 𝐵)))
2618, 25sylbid 240 . . 3 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → (2o𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵)))
2715, 26mpd 15 . 2 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → (2oo 𝐵) ⊆ (𝐴o 𝐵))
2810, 27sstrd 4005 1 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  cdif 3959  wss 3962  c0 4338  {cpr 4632  Oncon0 6385  suc csuc 6387  (class class class)co 7430  1oc1o 8497  2oc2o 8498  o coe 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-oexp 8510
This theorem is referenced by: (None)
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