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Theorem nnoeomeqom 41897
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 483 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ ω)
2 nnon 7845 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
31, 2syl 17 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ On)
4 omelon 9625 . . . . 5 ω ∈ On
5 limom 7855 . . . . 5 Lim ω
64, 5pm3.2i 471 . . . 4 (ω ∈ On ∧ Lim ω)
76a1i 11 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → (ω ∈ On ∧ Lim ω))
8 0elon 6408 . . . . 5 ∅ ∈ On
98a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ On)
10 0ss 4393 . . . . 5 ∅ ⊆ 1o
1110a1i 11 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ⊆ 1o)
12 simpr 485 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → 1o𝐴)
13 ontr2 6401 . . . . 5 ((∅ ∈ On ∧ 𝐴 ∈ On) → ((∅ ⊆ 1o ∧ 1o𝐴) → ∅ ∈ 𝐴))
1413imp 407 . . . 4 (((∅ ∈ On ∧ 𝐴 ∈ On) ∧ (∅ ⊆ 1o ∧ 1o𝐴)) → ∅ ∈ 𝐴)
159, 3, 11, 12, 14syl22anc 837 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → ∅ ∈ 𝐴)
16 oelim 8518 . . 3 (((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝐴) → (𝐴o ω) = 𝑥 ∈ ω (𝐴o 𝑥))
173, 7, 15, 16syl21anc 836 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = 𝑥 ∈ ω (𝐴o 𝑥))
18 ovex 7427 . . . 4 (𝐴o 𝑥) ∈ V
1918dfiun2 5030 . . 3 𝑥 ∈ ω (𝐴o 𝑥) = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)}
20 eluniab 4917 . . . . . 6 (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)))
21 19.42v 1957 . . . . . . . 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 1850 . . . . . . . 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 303 . . . . . . 7 ((𝑧𝑦 ∧ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑥(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
2726exbii 1850 . . . . . 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 296 . . . . 5 (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ ∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
30 simpr3 1196 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑦 = (𝐴o 𝑥))
31 simp2 1137 . . . . . . . . . . . 12 ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑥 ∈ ω)
32 nnecl 8598 . . . . . . . . . . . 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 6396 . . . . . . . . . . 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 4017 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑦 ⊆ ω)
37 simpr1 1194 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑧𝑦)
3836, 37sseldd 3980 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ (𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥))) → 𝑧 ∈ ω)
3938ex 413 . . . . . . 7 ((𝐴 ∈ ω ∧ 1o𝐴) → ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑧 ∈ ω))
4039exlimdvv 1937 . . . . . 6 ((𝐴 ∈ ω ∧ 1o𝐴) → (∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) → 𝑧 ∈ ω))
41 peano2 7865 . . . . . . . . 9 (𝑧 ∈ ω → suc 𝑧 ∈ ω)
4241adantl 482 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → suc 𝑧 ∈ ω)
43 ovex 7427 . . . . . . . . . 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 8486 . . . . . . . . . . . . 13 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
4745, 46sylibr 233 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝐴 ∈ (On ∖ 2o))
48 nnon 7845 . . . . . . . . . . . . 13 (suc 𝑧 ∈ ω → suc 𝑧 ∈ On)
4941, 48syl 17 . . . . . . . . . . . 12 (𝑧 ∈ ω → suc 𝑧 ∈ On)
50 oeworde 8578 . . . . . . . . . . . 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 3478 . . . . . . . . . . . . 13 𝑧 ∈ V
5352sucid 6436 . . . . . . . . . . . 12 𝑧 ∈ suc 𝑧
5453a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ suc 𝑧)
5551, 54sseldd 3980 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → 𝑧 ∈ (𝐴o suc 𝑧))
56 eqidd 2733 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝐴o suc 𝑧) = (𝐴o suc 𝑧))
5755, 42, 563jca 1128 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → (𝑧 ∈ (𝐴o suc 𝑧) ∧ suc 𝑧 ∈ ω ∧ (𝐴o suc 𝑧) = (𝐴o suc 𝑧)))
58 eleq2 2822 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑧𝑦𝑧 ∈ (𝐴o suc 𝑧)))
59 eqeq1 2736 . . . . . . . . . 10 (𝑦 = (𝐴o suc 𝑧) → (𝑦 = (𝐴o suc 𝑧) ↔ (𝐴o suc 𝑧) = (𝐴o suc 𝑧)))
6058, 593anbi13d 1438 . . . . . . . . 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 3586 . . . . . . . 8 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧)))
62 eleq1 2821 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑥 ∈ ω ↔ suc 𝑧 ∈ ω))
63 oveq2 7402 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (𝐴o 𝑥) = (𝐴o suc 𝑧))
6463eqeq2d 2743 . . . . . . . . . 10 (𝑥 = suc 𝑧 → (𝑦 = (𝐴o 𝑥) ↔ 𝑦 = (𝐴o suc 𝑧)))
6562, 643anbi23d 1439 . . . . . . . . 9 (𝑥 = suc 𝑧 → ((𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ (𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6665exbidv 1924 . . . . . . . 8 (𝑥 = suc 𝑧 → (∃𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)) ↔ ∃𝑦(𝑧𝑦 ∧ suc 𝑧 ∈ ω ∧ 𝑦 = (𝐴o suc 𝑧))))
6742, 61, 66spcedv 3586 . . . . . . 7 (((𝐴 ∈ ω ∧ 1o𝐴) ∧ 𝑧 ∈ ω) → ∃𝑥𝑦(𝑧𝑦𝑥 ∈ ω ∧ 𝑦 = (𝐴o 𝑥)))
6867ex 413 . . . . . 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 282 . . . 4 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝑧 {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} ↔ 𝑧 ∈ ω))
7170eqrdv 2730 . . 3 ((𝐴 ∈ ω ∧ 1o𝐴) → {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 = (𝐴o 𝑥)} = ω)
7219, 71eqtrid 2784 . 2 ((𝐴 ∈ ω ∧ 1o𝐴) → 𝑥 ∈ ω (𝐴o 𝑥) = ω)
7317, 72eqtrd 2772 1 ((𝐴 ∈ ω ∧ 1o𝐴) → (𝐴o ω) = ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wrex 3070  Vcvv 3474  cdif 3942  wss 3945  c0 4319   cuni 4902   ciun 4991  Oncon0 6354  Lim wlim 6355  suc csuc 6356  (class class class)co 7394  ωcom 7839  1oc1o 8443  2oc2o 8444  o coe 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709  ax-inf2 9620
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-oadd 8454  df-omul 8455  df-oexp 8456
This theorem is referenced by:  oenord1ex  41900  oaomoencom  41902
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