Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oege2 Structured version   Visualization version   GIF version

Theorem oege2 43320
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 8631. (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 8520 . . . . 5 2o ∈ On
2 1oex 8516 . . . . . . 7 1o ∈ V
32prid2 4763 . . . . . 6 1o ∈ {∅, 1o}
4 df2o3 8514 . . . . . 6 2o = {∅, 1o}
53, 4eleqtrri 2840 . . . . 5 1o ∈ 2o
6 ondif2 8540 . . . . 5 (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
71, 5, 6mpbir2an 711 . . . 4 2o ∈ (On ∖ 2o)
8 oeworde 8631 . . . 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 8507 . . . 4 2o = suc 1o
12 onsucss 43279 . . . . . 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 4022 . . 3 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 2o𝐴)
16 simpll 767 . . . . 5 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
17 onsseleq 6425 . . . . 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 8630 . . . . . 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 7438 . . . . . . 7 (2o = 𝐴 → (2oo 𝐵) = (𝐴o 𝐵))
22 ssid 4006 . . . . . . 7 (𝐴o 𝐵) ⊆ (𝐴o 𝐵)
2321, 22eqsstrdi 4028 . . . . . 6 (2o = 𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵))
2423a1i 11 . . . . 5 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → (2o = 𝐴 → (2oo 𝐵) ⊆ (𝐴o 𝐵)))
2520, 24jaod 860 . . . 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 3994 1 (((𝐴 ∈ On ∧ 1o𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  cdif 3948  wss 3951  c0 4333  {cpr 4628  Oncon0 6384  suc csuc 6386  (class class class)co 7431  1oc1o 8499  2oc2o 8500  o coe 8505
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator