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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 3949 | . . 3 ⊢ (𝑥 ∈ (On ∖ Fin) ↔ (𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin)) | |
2 | omelon 9669 | . . . . . 6 ⊢ ω ∈ On | |
3 | ontri1 6398 | . . . . . . . . 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 354 | . . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 ↔ 𝑥 ∈ ω)) |
6 | nnfi 9190 | . . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
7 | 5, 6 | biimtrdi 252 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
8 | 2, 7 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ On → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
9 | 8 | con1d 145 | . . . 4 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥)) |
10 | 9 | imp 405 | . . 3 ⊢ ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥) |
11 | 1, 10 | sylbi 216 | . 2 ⊢ (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥) |
12 | 11 | rgen 3053 | 1 ⊢ ∀𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ∀wral 3051 ∖ cdif 3936 ⊆ wss 3939 Oncon0 6364 ωcom 7868 Fincfn 8962 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 ax-inf2 9664 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-om 7869 df-en 8963 df-fin 8966 |
This theorem is referenced by: (None) |
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