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Theorem nnoeomeqom 42365
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 7865 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
31, 2syl 17 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ On)
4 omelon 9645 . . . . 5 ω ∈ On
5 limom 7875 . . . . 5 Lim ω
64, 5pm3.2i 470 . . . 4 (ω ∈ On ∧ Lim ω)
76a1i 11 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → (ω ∈ On ∧ Lim ω))
8 0elon 6418 . . . . 5 ∅ ∈ On
98a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ On)
10 0ss 4396 . . . . 5 ∅ ⊆ 1o
1110a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ⊆ 1o)
12 simpr 484 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → 1o𝐴)
13 ontr2 6411 . . . . 5 ((∅ ∈ On ∧ 𝐴 ∈ On) → ((∅ ⊆ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
1413imp 406 . . . 4 (((∅ ∈ On ∧ 𝐴 ∈ On) ∧ (∅ ⊆ 1o ∧ 1o𝐴)) → ∅ ∈ 𝐴)
159, 3, 11, 12, 14syl22anc 836 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ 𝐴)
16 oelim 8538 . . 3 (((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝐴) → (𝐴o ω) = 𝑥 ∈ ω (𝐴o 𝑥))
173, 7, 15, 16syl21anc 835 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = 𝑥 ∈ ω (𝐴o 𝑥))
18 ovex 7445 . . . 4 (𝐴o 𝑥) ∈ V
1918dfiun2 5036 . . 3 𝑥 ∈ ω (𝐴o 𝑥) = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)}
20 eluniab 4923 . . . . . 6 (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)))
21 19.42v 1956 . . . . . . . 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 1849 . . . . . . . 8 (∃𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑥(𝑧𝑦 ∧ (𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))))
24 df-rex 3070 . . . . . . . . 9 (∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥) ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
2524anbi2i 622 . . . . . . . 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 1849 . . . . . 6 (∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
28 excom 2161 . . . . . 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 8617 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴o 𝑥) ∈ ω)
331, 31, 32syl2an 595 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → (𝐴o 𝑥) ∈ ω)
34 onelss 6406 . . . . . . . . . . 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 4020 . . . . . . . . 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 3983 . . . . . . . 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 1936 . . . . . 6 ((𝐴 ∈ ω ∧ 1o𝐴) → (∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑧 ∈ ω))
41 peano2 7885 . . . . . . . . 9 (𝑧 ∈ ω → suc 𝑧 ∈ ω)
4241adantl 481 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ∈ ω)
43 ovex 7445 . . . . . . . . . 10 (𝐴o suc 𝑧) ∈ V
4443a1i 11 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝐴o suc 𝑧) ∈ V)
452anim1i 614 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴 ∈ On ∧ 1o𝐴))
46 ondif2 8506 . . . . . . . . . . . . 13 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
4745, 46sylibr 233 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ (On ∖ 2o))
48 nnon 7865 . . . . . . . . . . . . 13 (suc 𝑧 ∈ ω → suc 𝑧 ∈ On)
4941, 48syl 17 . . . . . . . . . . . 12 (𝑧 ∈ ω → suc 𝑧 ∈ On)
50 oeworde 8597 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝑧 ∈ On) → suc 𝑧 ⊆ (𝐴o suc 𝑧))
5147, 49, 50syl2an 595 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ⊆ (𝐴o suc 𝑧))
52 vex 3477 . . . . . . . . . . . . 13 𝑧 ∈ V
5352sucid 6446 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
5453a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ suc 𝑧)
5551, 54sseldd 3983 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ (𝐴o suc 𝑧))
56 eqidd 2732 . . . . . . . . . 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 2821 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑧𝑦𝑧 ∈ (𝐴o suc 𝑧)))
59 eqeq1 2735 . . . . . . . . . 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 3588 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧)))
62 eleq1 2820 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω))
63 oveq2 7420 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (𝐴o 𝑥) = (𝐴o suc 𝑧))
6463eqeq2d 2742 . . . . . . . . . 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 1923 . . . . . . . 8 (𝑥 = suc 𝑧 → (∃𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6742, 61, 66spcedv 3588 . . . . . . 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 211 . . . . 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 2729 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} = ω)
7219, 71eqtrid 2783 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝑥 ∈ ω (𝐴o 𝑥) = ω)
7317, 72eqtrd 2771 1 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wrex 3069  Vcvv 3473  cdif 3945  wss 3948  c0 4322   cuni 4908   ciun 4997  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7412  ωcom 7859  1oc1o 8463  2oc2o 8464  o coe 8469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-inf2 9640
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-omul 8475  df-oexp 8476
This theorem is referenced by:  oenord1ex  42368  oaomoencom  42370
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