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Theorem onexomgt 43258
Description: For any ordinal, there is always a larger product of omega. (Contributed by RP, 1-Feb-2025.)
Assertion
Ref Expression
onexomgt (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem onexomgt
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omelon 9687 . . 3 ω ∈ On
2 peano1 7911 . . . 4 ∅ ∈ ω
32ne0ii 4343 . . 3 ω ≠ ∅
4 omeu 8624 . . 3 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅) → ∃!𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
51, 3, 4mp3an13 1453 . 2 (𝐴 ∈ On → ∃!𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
6 euex 2576 . . 3 (∃!𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
7 onsuc 7832 . . . . . . . . . 10 (𝑎 ∈ On → suc 𝑎 ∈ On)
87adantr 480 . . . . . . . . 9 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → suc 𝑎 ∈ On)
9 simpr 484 . . . . . . . . . 10 (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) = 𝐴)
10 simpr 484 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑏 ∈ ω)
11 simpl 482 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
12 omcl 8575 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
131, 11, 12sylancr 587 . . . . . . . . . . . . . 14 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o 𝑎) ∈ On)
14 oaordi 8585 . . . . . . . . . . . . . 14 ((ω ∈ On ∧ (ω ·o 𝑎) ∈ On) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
151, 13, 14sylancr 587 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
1610, 15mpd 15 . . . . . . . . . . . 12 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω))
17 omsuc 8565 . . . . . . . . . . . . 13 ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
181, 11, 17sylancr 587 . . . . . . . . . . . 12 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
1916, 18eleqtrrd 2843 . . . . . . . . . . 11 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
2019adantr 480 . . . . . . . . . 10 (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
219, 20eqeltrrd 2841 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc 𝑎))
22 oveq2 7440 . . . . . . . . . . 11 (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω ·o suc 𝑎))
2322eleq2d 2826 . . . . . . . . . 10 (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc 𝑎)))
2423rspcev 3621 . . . . . . . . 9 ((suc 𝑎 ∈ On ∧ 𝐴 ∈ (ω ·o suc 𝑎)) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
258, 21, 24syl2an2r 685 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
2625ex 412 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (((ω ·o 𝑎) +o 𝑏) = 𝐴 → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
2726adantld 490 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
2827a1i 11 . . . . 5 (𝐴 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))))
2928rexlimdvv 3211 . . . 4 (𝐴 ∈ On → (∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
3029exlimdv 1932 . . 3 (𝐴 ∈ On → (∃𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
316, 30syl5 34 . 2 (𝐴 ∈ On → (∃!𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
325, 31mpd 15 1 (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  ∃!weu 2567  wne 2939  wrex 3069  c0 4332  cop 4631  Oncon0 6383  suc csuc 6385  (class class class)co 7432  ωcom 7888   +o coa 8504   ·o comu 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512
This theorem is referenced by:  onexlimgt  43260
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