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Theorem nnoeomeqom 43325
Description: Any natural number at least as large as two raised to the power of omega is omega. Lemma 3.25 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
Assertion
Ref Expression
nnoeomeqom ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = ω)

Proof of Theorem nnoeomeqom
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ ω)
2 nnon 7893 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
31, 2syl 17 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ On)
4 omelon 9686 . . . . 5 ω ∈ On
5 limom 7903 . . . . 5 Lim ω
64, 5pm3.2i 470 . . . 4 (ω ∈ On ∧ Lim ω)
76a1i 11 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → (ω ∈ On ∧ Lim ω))
8 0elon 6438 . . . . 5 ∅ ∈ On
98a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ On)
10 0ss 4400 . . . . 5 ∅ ⊆ 1o
1110a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ⊆ 1o)
12 simpr 484 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → 1o𝐴)
13 ontr2 6431 . . . . 5 ((∅ ∈ On ∧ 𝐴 ∈ On) → ((∅ ⊆ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
1413imp 406 . . . 4 (((∅ ∈ On ∧ 𝐴 ∈ On) ∧ (∅ ⊆ 1o ∧ 1o𝐴)) → ∅ ∈ 𝐴)
159, 3, 11, 12, 14syl22anc 839 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ 𝐴)
16 oelim 8572 . . 3 (((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝐴) → (𝐴o ω) = 𝑥 ∈ ω (𝐴o 𝑥))
173, 7, 15, 16syl21anc 838 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = 𝑥 ∈ ω (𝐴o 𝑥))
18 ovex 7464 . . . 4 (𝐴o 𝑥) ∈ V
1918dfiun2 5033 . . 3 𝑥 ∈ ω (𝐴o 𝑥) = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)}
20 eluniab 4921 . . . . . 6 (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)))
21 19.42v 1953 . . . . . . . 8 (∃𝑥(𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) ↔ (𝑧𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
22 3anass 1095 . . . . . . . . 9 ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ (𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
2322exbii 1848 . . . . . . . 8 (∃𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑥(𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
24 df-rex 3071 . . . . . . . . 9 (∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥) ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
2524anbi2i 623 . . . . . . . 8 ((𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)) ↔ (𝑧𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
2621, 23, 253bitr4ri 304 . . . . . . 7 ((𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
2726exbii 1848 . . . . . 6 (∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
28 excom 2162 . . . . . 6 (∃𝑦𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
2920, 27, 283bitri 297 . . . . 5 (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ ∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
30 simpr3 1197 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑦 = (𝐴o 𝑥))
31 simp2 1138 . . . . . . . . . . . 12 ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑥 ∈ ω)
32 nnecl 8651 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴o 𝑥) ∈ ω)
331, 31, 32syl2an 596 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → (𝐴o 𝑥) ∈ ω)
34 onelss 6426 . . . . . . . . . . 11 (ω ∈ On → ((𝐴o 𝑥) ∈ ω → (𝐴o 𝑥) ⊆ ω))
354, 33, 34mpsyl 68 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → (𝐴o 𝑥) ⊆ ω)
3630, 35eqsstrd 4018 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑦 ⊆ ω)
37 simpr1 1195 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑧𝑦)
3836, 37sseldd 3984 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑧 ∈ ω)
3938ex 412 . . . . . . 7 ((𝐴 ∈ ω ∧ 1o𝐴) → ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑧 ∈ ω))
4039exlimdvv 1934 . . . . . 6 ((𝐴 ∈ ω ∧ 1o𝐴) → (∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑧 ∈ ω))
41 peano2 7912 . . . . . . . . 9 (𝑧 ∈ ω → suc 𝑧 ∈ ω)
4241adantl 481 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ∈ ω)
43 ovex 7464 . . . . . . . . . 10 (𝐴o suc 𝑧) ∈ V
4443a1i 11 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝐴o suc 𝑧) ∈ V)
452anim1i 615 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴 ∈ On ∧ 1o𝐴))
46 ondif2 8540 . . . . . . . . . . . . 13 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
4745, 46sylibr 234 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ (On ∖ 2o))
48 nnon 7893 . . . . . . . . . . . . 13 (suc 𝑧 ∈ ω → suc 𝑧 ∈ On)
4941, 48syl 17 . . . . . . . . . . . 12 (𝑧 ∈ ω → suc 𝑧 ∈ On)
50 oeworde 8631 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝑧 ∈ On) → suc 𝑧 ⊆ (𝐴o suc 𝑧))
5147, 49, 50syl2an 596 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ⊆ (𝐴o suc 𝑧))
52 vex 3484 . . . . . . . . . . . . 13 𝑧 ∈ V
5352sucid 6466 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
5453a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ suc 𝑧)
5551, 54sseldd 3984 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ (𝐴o suc 𝑧))
56 eqidd 2738 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝐴o suc 𝑧) = (𝐴o suc 𝑧))
5755, 42, 563jca 1129 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝑧 ∈ (𝐴o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴o suc 𝑧) = (𝐴o suc 𝑧)))
58 eleq2 2830 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑧𝑦𝑧 ∈ (𝐴o suc 𝑧)))
59 eqeq1 2741 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑦 = (𝐴o suc 𝑧) ↔ (𝐴o suc 𝑧) = (𝐴o suc 𝑧)))
6058, 593anbi13d 1440 . . . . . . . . 9 (𝑦 = (𝐴o suc 𝑧) → ((𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧)) ↔ (𝑧 ∈ (𝐴o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴o suc 𝑧) = (𝐴o suc 𝑧))))
6144, 57, 60spcedv 3598 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧)))
62 eleq1 2829 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω))
63 oveq2 7439 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (𝐴o 𝑥) = (𝐴o suc 𝑧))
6463eqeq2d 2748 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑦 = (𝐴o 𝑥) ↔ 𝑦 = (𝐴o suc 𝑧)))
6562, 643anbi23d 1441 . . . . . . . . 9 (𝑥 = suc 𝑧 → ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ (𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6665exbidv 1921 . . . . . . . 8 (𝑥 = suc 𝑧 → (∃𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6742, 61, 66spcedv 3598 . . . . . . 7 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → ∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
6867ex 412 . . . . . 6 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝑧 ∈ ω → ∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
6940, 68impbid 212 . . . . 5 ((𝐴 ∈ ω ∧ 1o𝐴) → (∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ 𝑧 ∈ ω))
7029, 69bitrid 283 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ 𝑧 ∈ ω))
7170eqrdv 2735 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} = ω)
7219, 71eqtrid 2789 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝑥 ∈ ω (𝐴o 𝑥) = ω)
7317, 72eqtrd 2777 1 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  c0 4333   cuni 4907   ciun 4991  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  ωcom 7887  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  ax-inf2 9681
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:  oenord1ex  43328  oaomoencom  43330
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