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Mirrors > Home > MPE Home > Th. List > Mathboxes > infordmin | Structured version Visualization version GIF version |
Description: ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.) |
Ref | Expression |
---|---|
infordmin | ⊢ ∀𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3946 | . . 3 ⊢ (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin)) | |
2 | omelon 9109 | . . . . . 6 ⊢ ω ∈ On | |
3 | ontri1 6225 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω)) | |
4 | 3 | bicomd 225 | . . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥)) |
5 | 4 | con1bid 358 | . . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 ↔ 𝑥 ∈ ω)) |
6 | nnfi 8711 | . . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
7 | 5, 6 | syl6bi 255 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
8 | 2, 7 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ On → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
9 | 8 | con1d 147 | . . . 4 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥)) |
10 | 9 | imp 409 | . . 3 ⊢ ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥) |
11 | 1, 10 | sylbi 219 | . 2 ⊢ (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥) |
12 | 11 | rgen 3148 | 1 ⊢ ∀𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 ∖ cdif 3933 ⊆ wss 3936 Oncon0 6191 ωcom 7580 Fincfn 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 |
This theorem is referenced by: (None) |
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