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Mirrors > Home > MPE Home > Th. List > 1sdom2 | Structured version Visualization version GIF version |
Description: Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un 7677, see 1sdom2ALT 9192. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7677. (Revised by BTernaryTau, 8-Dec-2024.) |
Ref | Expression |
---|---|
1sdom2 | ⊢ 1o ≺ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 8433 | . . . 4 ⊢ 2o ≠ ∅ | |
2 | 2oex 8428 | . . . . 5 ⊢ 2o ∈ V | |
3 | 2 | 0sdom 9058 | . . . 4 ⊢ (∅ ≺ 2o ↔ 2o ≠ ∅) |
4 | 1, 3 | mpbir 230 | . . 3 ⊢ ∅ ≺ 2o |
5 | 0sdom1dom 9189 | . . 3 ⊢ (∅ ≺ 2o ↔ 1o ≼ 2o) | |
6 | 4, 5 | mpbi 229 | . 2 ⊢ 1o ≼ 2o |
7 | snnen2o 9188 | . . 3 ⊢ ¬ {∅} ≈ 2o | |
8 | df1o2 8424 | . . . 4 ⊢ 1o = {∅} | |
9 | 8 | breq1i 5117 | . . 3 ⊢ (1o ≈ 2o ↔ {∅} ≈ 2o) |
10 | 7, 9 | mtbir 323 | . 2 ⊢ ¬ 1o ≈ 2o |
11 | brsdom 8922 | . 2 ⊢ (1o ≺ 2o ↔ (1o ≼ 2o ∧ ¬ 1o ≈ 2o)) | |
12 | 6, 10, 11 | mpbir2an 710 | 1 ⊢ 1o ≺ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ≠ wne 2944 ∅c0 4287 {csn 4591 class class class wbr 5110 1oc1o 8410 2oc2o 8411 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1o 8417 df-2o 8418 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: pm54.43 9944 pr2neOLD 9948 prdom2 9949 canthp1lem1 10595 canthp1 10597 1nprm 16562 |
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