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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 a hypothesis. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
nnsdomg | ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdomg 8548 | . . 3 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω)) | |
2 | ordom 7582 | . . . 4 ⊢ Ord ω | |
3 | ordelss 6200 | . . . 4 ⊢ ((Ord ω ∧ 𝐴 ∈ ω) → 𝐴 ⊆ ω) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) |
5 | 1, 4 | impel 508 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≼ ω) |
6 | ominf 8723 | . . . 4 ⊢ ¬ ω ∈ Fin | |
7 | ensym 8551 | . . . . 5 ⊢ (𝐴 ≈ ω → ω ≈ 𝐴) | |
8 | breq2 5063 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (ω ≈ 𝑥 ↔ ω ≈ 𝐴)) | |
9 | 8 | rspcev 3620 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ∃𝑥 ∈ ω ω ≈ 𝑥) |
10 | isfi 8526 | . . . . . . 7 ⊢ (ω ∈ Fin ↔ ∃𝑥 ∈ ω ω ≈ 𝑥) | |
11 | 9, 10 | sylibr 236 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ω ≈ 𝐴) → ω ∈ Fin) |
12 | 11 | ex 415 | . . . . 5 ⊢ (𝐴 ∈ ω → (ω ≈ 𝐴 → ω ∈ Fin)) |
13 | 7, 12 | syl5 34 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ≈ ω → ω ∈ Fin)) |
14 | 6, 13 | mtoi 201 | . . 3 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ ω) |
15 | 14 | adantl 484 | . 2 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → ¬ 𝐴 ≈ ω) |
16 | brsdom 8525 | . 2 ⊢ (𝐴 ≺ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≈ ω)) | |
17 | 5, 15, 16 | sylanbrc 585 | 1 ⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2113 ∃wrex 3138 Vcvv 3491 ⊆ wss 3929 class class class wbr 5059 Ord word 6183 ωcom 7573 ≈ cen 8499 ≼ cdom 8500 ≺ csdm 8501 Fincfn 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7574 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 |
This theorem is referenced by: isfiniteg 8771 infsdomnn 8772 nnsdom 9110 |
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