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Mirrors > Home > MPE Home > Th. List > infsdomnn | Structured version Visualization version GIF version |
Description: An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5371. (Revised by BTernaryTau, 7-Jan-2025.) |
Ref | Expression |
---|---|
infsdomnn | ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnfi 9207 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ Fin) | |
2 | 1 | adantl 480 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ∈ Fin) |
3 | reldom 8982 | . . . 4 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5740 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ V) |
5 | nnsdomg 9338 | . . 3 ⊢ ((ω ∈ V ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) | |
6 | 4, 5 | sylan 578 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ ω) |
7 | simpl 481 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → ω ≼ 𝐴) | |
8 | sdomdomtrfi 9240 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≺ ω ∧ ω ≼ 𝐴) → 𝐵 ≺ 𝐴) | |
9 | 2, 6, 7, 8 | syl3anc 1368 | 1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 Vcvv 3462 class class class wbr 5155 ωcom 7878 ≼ cdom 8974 ≺ csdm 8975 Fincfn 8976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-om 7879 df-1o 8498 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 |
This theorem is referenced by: infn0ALT 9344 |
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