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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 7587 | . . 3 ⊢ ∅ ∈ ω | |
2 | infsdomnn 8765 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ ∅ ∈ ω) → ∅ ≺ 𝐴) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (ω ≼ 𝐴 → ∅ ≺ 𝐴) |
4 | reldom 8501 | . . . 4 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5595 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | 0sdomg 8632 | . . 3 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (ω ≼ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 3, 7 | mpbid 234 | 1 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ≠ wne 3016 Vcvv 3486 ∅c0 4279 class class class wbr 5052 ωcom 7566 ≼ cdom 8493 ≺ csdm 8494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-om 7567 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 |
This theorem is referenced by: infpwfien 9474 infxp 9623 infpss 9625 alephmul 9986 csdfil 22485 |
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