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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 7729, see 1sdom2ALT 9247. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7729. (Revised by BTernaryTau, 8-Dec-2024.) |
Ref | Expression |
---|---|
1sdom2 | ⊢ 1o ≺ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 8488 | . . . 4 ⊢ 2o ≠ ∅ | |
2 | 2oex 8483 | . . . . 5 ⊢ 2o ∈ V | |
3 | 2 | 0sdom 9113 | . . . 4 ⊢ (∅ ≺ 2o ↔ 2o ≠ ∅) |
4 | 1, 3 | mpbir 230 | . . 3 ⊢ ∅ ≺ 2o |
5 | 0sdom1dom 9244 | . . 3 ⊢ (∅ ≺ 2o ↔ 1o ≼ 2o) | |
6 | 4, 5 | mpbi 229 | . 2 ⊢ 1o ≼ 2o |
7 | snnen2o 9243 | . . 3 ⊢ ¬ {∅} ≈ 2o | |
8 | df1o2 8479 | . . . 4 ⊢ 1o = {∅} | |
9 | 8 | breq1i 5155 | . . 3 ⊢ (1o ≈ 2o ↔ {∅} ≈ 2o) |
10 | 7, 9 | mtbir 323 | . 2 ⊢ ¬ 1o ≈ 2o |
11 | brsdom 8977 | . 2 ⊢ (1o ≺ 2o ↔ (1o ≼ 2o ∧ ¬ 1o ≈ 2o)) | |
12 | 6, 10, 11 | mpbir2an 708 | 1 ⊢ 1o ≺ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ≠ wne 2939 ∅c0 4322 {csn 4628 class class class wbr 5148 1oc1o 8465 2oc2o 8466 ≈ cen 8942 ≼ cdom 8943 ≺ csdm 8944 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1o 8472 df-2o 8473 df-en 8946 df-dom 8947 df-sdom 8948 |
This theorem is referenced by: pm54.43 10002 pr2neOLD 10006 prdom2 10007 canthp1lem1 10653 canthp1 10655 1nprm 16623 |
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