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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 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)) | |
4 | 3 | bicomd 222 | . . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ω ↔ ω ⊆ 𝑥)) |
5 | 4 | con1bid 356 | . . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 ↔ 𝑥 ∈ ω)) |
6 | nnfi 9112 | . . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
7 | 5, 6 | syl6bi 253 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
8 | 2, 7 | mpan 689 | . . . . 5 ⊢ (𝑥 ∈ On → (¬ ω ⊆ 𝑥 → 𝑥 ∈ Fin)) |
9 | 8 | con1d 145 | . . . 4 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ Fin → ω ⊆ 𝑥)) |
10 | 9 | imp 408 | . . 3 ⊢ ((𝑥 ∈ On ∧ ¬ 𝑥 ∈ Fin) → ω ⊆ 𝑥) |
11 | 1, 10 | sylbi 216 | . 2 ⊢ (𝑥 ∈ (On ∖ Fin) → ω ⊆ 𝑥) |
12 | 11 | rgen 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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