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Mirrors > Home > MPE Home > Th. List > 0sdom1dom | Structured version Visualization version GIF version |
Description: Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.) |
Ref | Expression |
---|---|
0sdom1dom | ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7620 | . . 3 ⊢ ∅ ∈ ω | |
2 | sucdom 8794 | . . 3 ⊢ (∅ ∈ ω → (∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴) |
4 | df-1o 8131 | . . 3 ⊢ 1o = suc ∅ | |
5 | 4 | breq1i 5037 | . 2 ⊢ (1o ≼ 𝐴 ↔ suc ∅ ≼ 𝐴) |
6 | 3, 5 | bitr4i 281 | 1 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2114 ∅c0 4211 class class class wbr 5030 suc csuc 6174 ωcom 7599 1oc1o 8124 ≼ cdom 8553 ≺ csdm 8554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7600 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 |
This theorem is referenced by: sdom1 8797 djulepw 9692 fin45 9892 gchxpidm 10169 rankcf 10277 snct 30623 |
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