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Theorem infordmin 41811
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 3921 . . 3 (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin))
2 omelon 9583 . . . . . 6 ω ∈ On
3 ontri1 6352 . . . . . . . . 9 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
43bicomd 222 . . . . . . . 8 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥))
54con1bid 356 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ ω))
6 nnfi 9112 . . . . . . 7 (𝑥 ∈ ω → 𝑥 ∈ Fin)
75, 6syl6bi 253 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
82, 7mpan 689 . . . . 5 (𝑥 ∈ On → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
98con1d 145 . . . 4 (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥))
109imp 408 . . 3 ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥)
111, 10sylbi 216 . 2 (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥)
1211rgen 3067 1 𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  wral 3065  cdif 3908  wss 3911  Oncon0 6318  ωcom 7803  Fincfn 8884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-inf2 9578
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-om 7804  df-en 8885  df-fin 8888
This theorem is referenced by: (None)
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