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Theorem infordmin 40096
 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 3929 . . 3 (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin))
2 omelon 9102 . . . . . 6 ω ∈ On
3 ontri1 6213 . . . . . . . . 9 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
43bicomd 226 . . . . . . . 8 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥))
54con1bid 359 . . . . . . 7 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ ω))
6 nnfi 8705 . . . . . . 7 (𝑥 ∈ ω → 𝑥 ∈ Fin)
75, 6syl6bi 256 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
82, 7mpan 689 . . . . 5 (𝑥 ∈ On → (¬ ω ⊆ 𝑥𝑥 ∈ Fin))
98con1d 147 . . . 4 (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥))
109imp 410 . . 3 ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥)
111, 10sylbi 220 . 2 (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥)
1211rgen 3143 1 𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2115  ∀wral 3133   ∖ cdif 3916   ⊆ wss 3919  Oncon0 6179  ωcom 7571  Fincfn 8501 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-inf2 9097 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-om 7572  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505 This theorem is referenced by: (None)
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