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| Mirrors > Home > MPE Home > Th. List > Mathboxes > omnord1ex | Structured version Visualization version GIF version | ||
| Description: When omega is multiplied on the right to ordinals one and two, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 29-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| omnord1ex | ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1oex 8517 | . . . . 5 ⊢ 1o ∈ V | |
| 2 | 1 | prid2 4762 | . . . 4 ⊢ 1o ∈ {∅, 1o} | 
| 3 | df2o3 8515 | . . . 4 ⊢ 2o = {∅, 1o} | |
| 4 | 2, 3 | eleqtrri 2839 | . . 3 ⊢ 1o ∈ 2o | 
| 5 | ordom 7898 | . . . 4 ⊢ Ord ω | |
| 6 | ordirr 6401 | . . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω) | |
| 7 | omelon 9687 | . . . . . . 7 ⊢ ω ∈ On | |
| 8 | 1onn 8679 | . . . . . . 7 ⊢ 1o ∈ ω | |
| 9 | 0lt1o 8543 | . . . . . . 7 ⊢ ∅ ∈ 1o | |
| 10 | omabslem 8689 | . . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω) | |
| 11 | 7, 8, 9, 10 | mp3an 1462 | . . . . . 6 ⊢ (1o ·o ω) = ω | 
| 12 | 2omomeqom 43321 | . . . . . 6 ⊢ (2o ·o ω) = ω | |
| 13 | 11, 12 | eleq12i 2833 | . . . . 5 ⊢ ((1o ·o ω) ∈ (2o ·o ω) ↔ ω ∈ ω) | 
| 14 | 6, 13 | sylnibr 329 | . . . 4 ⊢ (Ord ω → ¬ (1o ·o ω) ∈ (2o ·o ω)) | 
| 15 | 5, 14 | ax-mp 5 | . . 3 ⊢ ¬ (1o ·o ω) ∈ (2o ·o ω) | 
| 16 | 4, 15 | 2th 264 | . 2 ⊢ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω)) | 
| 17 | xor3 382 | . 2 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) ↔ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω))) | |
| 18 | 16, 17 | mpbir 231 | 1 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∅c0 4332 {cpr 4627 Ord word 6382 Oncon0 6383 (class class class)co 7432 ωcom 7888 1oc1o 8500 2oc2o 8501 ·o comu 8505 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 | 
| This theorem is referenced by: omnord1 43323 | 
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