![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 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)) | |
4 | 3 | bicomd 222 | . . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥)) |
5 | 4 | con1bid 355 | . . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 ↔ 𝑥 ∈ ω)) |
6 | nnfi 9169 | . . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
7 | 5, 6 | syl6bi 253 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
8 | 2, 7 | mpan 687 | . . . . 5 ⊢ (𝑥 ∈ On → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
9 | 8 | con1d 145 | . . . 4 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥)) |
10 | 9 | imp 406 | . . 3 ⊢ ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥) |
11 | 1, 10 | sylbi 216 | . 2 ⊢ (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥) |
12 | 11 | rgen 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) |
Copyright terms: Public domain | W3C validator |