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Mirrors > Home > MPE Home > Th. List > peano1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of peano1 7872 as of 29-Nov-2024. (Contributed by NM, 15-May-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
peano1OLD | ⊢ ∅ ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limom 7864 | . 2 ⊢ Lim ω | |
2 | 0ellim 6417 | . 2 ⊢ (Lim ω → ∅ ∈ ω) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∅ ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∅c0 4314 Lim wlim 6355 ωcom 7848 |
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-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-tr 5256 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-om 7849 |
This theorem is referenced by: (None) |
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