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Mirrors > Home > MPE Home > Th. List > nnsdomg | Structured version Visualization version GIF version |
Description: Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of Infinity, we include it as part of the antecedent. See nnsdom 9648 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5356. (Revised by BTernaryTau, 7-Jan-2025.) |
Ref | Expression |
---|---|
nnsdomg | ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7861 | . . . . . 6 ⊢ Ord ω | |
2 | ordelss 6373 | . . . . . 6 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
3 | 1, 2 | mpan 687 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω) |
5 | nnfi 9166 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
6 | ssdomfi2 9199 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω) | |
7 | 5, 6 | syl3an1 1160 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω) |
8 | 4, 7 | mpd3an3 1458 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ≼ ω) |
9 | 8 | ancoms 458 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω) |
10 | ominf 9257 | . . . 4 ⊢ ¬ ω ∈ Fin | |
11 | ensymfib 9186 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴)) | |
12 | 5, 11 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴)) |
13 | breq2 5145 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴)) | |
14 | 13 | rspcev 3606 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥) |
15 | isfi 8971 | . . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
16 | 14, 15 | sylibr 233 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin) |
17 | 16 | ex 412 | . . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin)) |
18 | 12, 17 | sylbid 239 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin)) |
19 | 10, 18 | mtoi 198 | . . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω) |
20 | 19 | adantl 481 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω) |
21 | brsdom 8970 | . 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω)) | |
22 | 9, 20, 21 | sylanbrc 582 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 class class class wbr 5141 Ord word 6356 ωcom 7851 ≈ cen 8935 ≼ cdom 8936 ≺ csdm 8937 Fincfn 8938 |
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-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
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-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 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-mpt 5225 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-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7852 df-1o 8464 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 |
This theorem is referenced by: isfiniteg 9303 infsdomnn 9304 infsdomnnOLD 9305 nnsdom 9648 |
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