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Mirrors > Home > MPE Home > Th. List > infn0 | Structured version Visualization version GIF version |
Description: An infinite set is not empty. (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
infn0 | ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7347 | . . 3 ⊢ ∅ ∈ ω | |
2 | infsdomnn 8491 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ ∅ ∈ ω) → ∅ ≺ 𝐴) | |
3 | 1, 2 | mpan2 684 | . 2 ⊢ (ω ≼ 𝐴 → ∅ ≺ 𝐴) |
4 | reldom 8229 | . . . 4 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5395 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | 0sdomg 8359 | . . 3 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 3, 7 | mpbid 224 | 1 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2166 ≠ wne 3000 Vcvv 3415 ∅c0 4145 class class class wbr 4874 ωcom 7327 ≼ cdom 8221 ≺ csdm 8222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-om 7328 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 |
This theorem is referenced by: infpwfien 9199 cdainf 9330 infxp 9353 infpss 9355 alephmul 9716 csdfil 22069 |
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