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Theorem infordmin 42859
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 3953 . . 3 (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin))
2 omelon 9643 . . . . . 6 ω ∈ On
3 ontri1 6392 . . . . . . . . 9 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
43bicomd 222 . . . . . . . 8 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥))
54con1bid 355 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ ω))
6 nnfi 9169 . . . . . . 7 (𝑥 ∈ ω → 𝑥 ∈ Fin)
75, 6syl6bi 253 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
82, 7mpan 687 . . . . 5 (𝑥 ∈ On → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
98con1d 145 . . . 4 (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥))
109imp 406 . . 3 ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥)
111, 10sylbi 216 . 2 (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥)
1211rgen 3057 1 𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2098  wral 3055  cdif 3940  wss 3943  Oncon0 6358  ωcom 7852  Fincfn 8941
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-om 7853  df-en 8942  df-fin 8945
This theorem is referenced by: (None)
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