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Theorem infordmin 43521
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 3972 . . 3 (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin))
2 omelon 9683 . . . . . 6 ω ∈ On
3 ontri1 6419 . . . . . . . . 9 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
43bicomd 223 . . . . . . . 8 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥))
54con1bid 355 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ ω))
6 nnfi 9205 . . . . . . 7 (𝑥 ∈ ω → 𝑥 ∈ Fin)
75, 6biimtrdi 253 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
82, 7mpan 690 . . . . 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 217 . 2 (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥)
1211rgen 3060 1 𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wral 3058  cdif 3959  wss 3962  Oncon0 6385  ωcom 7886  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-om 7887  df-en 8984  df-fin 8987
This theorem is referenced by: (None)
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