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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 7721, see 1sdom2ALT 9240. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7721. (Revised by BTernaryTau, 8-Dec-2024.) |
Ref | Expression |
---|---|
1sdom2 | ⊢ 1o ≺ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 8480 | . . . 4 ⊢ 2o ≠ ∅ | |
2 | 2oex 8475 | . . . . 5 ⊢ 2o ∈ V | |
3 | 2 | 0sdom 9106 | . . . 4 ⊢ (∅ ≺ 2o ↔ 2o ≠ ∅) |
4 | 1, 3 | mpbir 230 | . . 3 ⊢ ∅ ≺ 2o |
5 | 0sdom1dom 9237 | . . 3 ⊢ (∅ ≺ 2o ↔ 1o ≼ 2o) | |
6 | 4, 5 | mpbi 229 | . 2 ⊢ 1o ≼ 2o |
7 | snnen2o 9236 | . . 3 ⊢ ¬ {∅} ≈ 2o | |
8 | df1o2 8471 | . . . 4 ⊢ 1o = {∅} | |
9 | 8 | breq1i 5148 | . . 3 ⊢ (1o ≈ 2o ↔ {∅} ≈ 2o) |
10 | 7, 9 | mtbir 323 | . 2 ⊢ ¬ 1o ≈ 2o |
11 | brsdom 8970 | . 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 2934 ∅c0 4317 {csn 4623 class class class wbr 5141 1oc1o 8457 2oc2o 8458 ≈ cen 8935 ≼ cdom 8936 ≺ csdm 8937 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-1o 8464 df-2o 8465 df-en 8939 df-dom 8940 df-sdom 8941 |
This theorem is referenced by: pm54.43 9995 pr2neOLD 9999 prdom2 10000 canthp1lem1 10646 canthp1 10648 1nprm 16621 |
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