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Theorem infordmin 43027
Description: ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.)
Assertion
Ref Expression
infordmin 𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥

Proof of Theorem infordmin
StepHypRef Expression
1 eldif 3949 . . 3 (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin))
2 omelon 9669 . . . . . 6 ω ∈ On
3 ontri1 6398 . . . . . . . . 9 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
43bicomd 222 . . . . . . . 8 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥))
54con1bid 354 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ ω))
6 nnfi 9190 . . . . . . 7 (𝑥 ∈ ω → 𝑥 ∈ Fin)
75, 6biimtrdi 252 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
82, 7mpan 688 . . . . 5 (𝑥 ∈ On → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
98con1d 145 . . . 4 (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥))
109imp 405 . . 3 ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥)
111, 10sylbi 216 . 2 (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥)
1211rgen 3053 1 𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wcel 2098  wral 3051  cdif 3936  wss 3939  Oncon0 6364  ωcom 7868  Fincfn 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-om 7869  df-en 8963  df-fin 8966
This theorem is referenced by: (None)
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