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Theorem nnoeomeqom 43301
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 7892 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
31, 2syl 17 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ On)
4 omelon 9683 . . . . 5 ω ∈ On
5 limom 7902 . . . . 5 Lim ω
64, 5pm3.2i 470 . . . 4 (ω ∈ On ∧ Lim ω)
76a1i 11 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → (ω ∈ On ∧ Lim ω))
8 0elon 6439 . . . . 5 ∅ ∈ On
98a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ On)
10 0ss 4405 . . . . 5 ∅ ⊆ 1o
1110a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ⊆ 1o)
12 simpr 484 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → 1o𝐴)
13 ontr2 6432 . . . . 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 8570 . . 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 7463 . . . 4 (𝐴o 𝑥) ∈ V
1918dfiun2 5037 . . 3 𝑥 ∈ ω (𝐴o 𝑥) = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)}
20 eluniab 4925 . . . . . 6 (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)))
21 19.42v 1950 . . . . . . . 8 (∃𝑥(𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) ↔ (𝑧𝑦 ∧ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
22 3anass 1094 . . . . . . . . 9 ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ (𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
2322exbii 1844 . . . . . . . 8 (∃𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑥(𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
24 df-rex 3068 . . . . . . . . 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 1844 . . . . . 6 (∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
28 excom 2159 . . . . . 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 1195 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑦 = (𝐴o 𝑥))
31 simp2 1136 . . . . . . . . . . . 12 ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑥 ∈ ω)
32 nnecl 8649 . . . . . . . . . . . 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 6427 . . . . . . . . . . 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 4033 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑦 ⊆ ω)
37 simpr1 1193 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑧𝑦)
3836, 37sseldd 3995 . . . . . . . 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 1931 . . . . . 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 7463 . . . . . . . . . 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 8538 . . . . . . . . . . . . 13 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
4745, 46sylibr 234 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ (On ∖ 2o))
48 nnon 7892 . . . . . . . . . . . . 13 (suc 𝑧 ∈ ω → suc 𝑧 ∈ On)
4941, 48syl 17 . . . . . . . . . . . 12 (𝑧 ∈ ω → suc 𝑧 ∈ On)
50 oeworde 8629 . . . . . . . . . . . 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 3481 . . . . . . . . . . . . 13 𝑧 ∈ V
5352sucid 6467 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
5453a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ suc 𝑧)
5551, 54sseldd 3995 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ (𝐴o suc 𝑧))
56 eqidd 2735 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝐴o suc 𝑧) = (𝐴o suc 𝑧))
5755, 42, 563jca 1127 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝑧 ∈ (𝐴o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴o suc 𝑧) = (𝐴o suc 𝑧)))
58 eleq2 2827 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑧𝑦𝑧 ∈ (𝐴o suc 𝑧)))
59 eqeq1 2738 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑦 = (𝐴o suc 𝑧) ↔ (𝐴o suc 𝑧) = (𝐴o suc 𝑧)))
6058, 593anbi13d 1437 . . . . . . . . 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 3597 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧)))
62 eleq1 2826 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω))
63 oveq2 7438 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (𝐴o 𝑥) = (𝐴o suc 𝑧))
6463eqeq2d 2745 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑦 = (𝐴o 𝑥) ↔ 𝑦 = (𝐴o suc 𝑧)))
6562, 643anbi23d 1438 . . . . . . . . 9 (𝑥 = suc 𝑧 → ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ (𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6665exbidv 1918 . . . . . . . 8 (𝑥 = suc 𝑧 → (∃𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6742, 61, 66spcedv 3597 . . . . . . 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 2732 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} = ω)
7219, 71eqtrid 2786 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝑥 ∈ ω (𝐴o 𝑥) = ω)
7317, 72eqtrd 2774 1 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wrex 3067  Vcvv 3477  cdif 3959  wss 3962  c0 4338   cuni 4911   ciun 4995  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430  ωcom 7886  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  ax-inf2 9678
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:  oenord1ex  43304  oaomoencom  43306
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