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Theorem onexomgt 43230
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 9684 . . 3 ω ∈ On
2 peano1 7911 . . . 4 ∅ ∈ ω
32ne0ii 4350 . . 3 ω ≠ ∅
4 omeu 8622 . . 3 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅) → ∃!𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
51, 3, 4mp3an13 1451 . 2 (𝐴 ∈ On → ∃!𝑐𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
6 euex 2575 . . 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 7831 . . . . . . . . . 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 8573 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
131, 11, 12sylancr 587 . . . . . . . . . . . . . 14 ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o 𝑎) ∈ On)
14 oaordi 8583 . . . . . . . . . . . . . 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 8563 . . . . . . . . . . . . 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 2842 . . . . . . . . . . 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 2840 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc 𝑎))
22 oveq2 7439 . . . . . . . . . . 11 (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω ·o suc 𝑎))
2322eleq2d 2825 . . . . . . . . . 10 (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc 𝑎)))
2423rspcev 3622 . . . . . . . . 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 3210 . . . 4 (𝐴 ∈ On → (∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
3029exlimdv 1931 . . 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 1537  wex 1776  wcel 2106  ∃!weu 2566  wne 2938  wrex 3068  c0 4339  cop 4637  Oncon0 6386  suc csuc 6388  (class class class)co 7431  ωcom 7887   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by:  onexlimgt  43232
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