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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 9685 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5369. (Revised by BTernaryTau, 7-Jan-2025.) |
Ref | Expression |
---|---|
nnsdomg | ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7886 | . . . . . 6 ⊢ Ord ω | |
2 | ordelss 6390 | . . . . . 6 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
3 | 1, 2 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
4 | 3 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ V) → 𝐴 ⊆ ω) |
5 | nnfi 9198 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
6 | ssdomfi2 9231 | . . . . 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 457 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω) |
10 | ominf 9289 | . . . 4 ⊢ ¬ ω ∈ Fin | |
11 | ensymfib 9218 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ ω ↔ ω ≈ 𝐴)) | |
12 | 5, 11 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω ↔ ω ≈ 𝐴)) |
13 | breq2 5156 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴)) | |
14 | 13 | rspcev 3611 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥) |
15 | isfi 9003 | . . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
16 | 14, 15 | sylibr 233 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin) |
17 | 16 | ex 411 | . . . . 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 480 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω) |
21 | brsdom 9002 | . 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω)) | |
22 | 9, 20, 21 | sylanbrc 581 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∃wrex 3067 Vcvv 3473 ⊆ wss 3949 class class class wbr 5152 Ord word 6373 ωcom 7876 ≈ cen 8967 ≼ cdom 8968 ≺ csdm 8969 Fincfn 8970 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 |
This theorem is referenced by: isfiniteg 9335 infsdomnn 9336 infsdomnnOLD 9337 nnsdom 9685 |
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