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Mirrors > Home > MPE Home > Th. List > negsproplem5 | Structured version Visualization version GIF version |
Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐵 is simpler than 𝐴. (Contributed by Scott Fenton, 3-Feb-2025.) |
Ref | Expression |
---|---|
negsproplem.1 | ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) |
negsproplem4.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
negsproplem4.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
negsproplem4.3 | ⊢ (𝜑 → 𝐴 <s 𝐵) |
negsproplem5.4 | ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) |
Ref | Expression |
---|---|
negsproplem5 | ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negsproplem.1 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) | |
2 | negsproplem4.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ No ) | |
3 | 1, 2 | negsproplem3 27955 | . . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)))) |
4 | 3 | simp2d 1140 | . 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)}) |
5 | negsfn 27949 | . . 3 ⊢ -us Fn No | |
6 | rightssno 27821 | . . 3 ⊢ ( R ‘𝐴) ⊆ No | |
7 | negsproplem5.4 | . . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) | |
8 | bdayelon 27722 | . . . . . 6 ⊢ ( bday ‘𝐴) ∈ On | |
9 | negsproplem4.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No ) | |
10 | oldbday 27840 | . . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴))) | |
11 | 8, 9, 10 | sylancr 585 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴))) |
12 | 7, 11 | mpbird 256 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ( O ‘( bday ‘𝐴))) |
13 | negsproplem4.3 | . . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
14 | breq2 5148 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴 <s 𝑏 ↔ 𝐴 <s 𝐵)) | |
15 | rightval 27804 | . . . . 5 ⊢ ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑏} | |
16 | 14, 15 | elrab2 3679 | . . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵)) |
17 | 12, 13, 16 | sylanbrc 581 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴)) |
18 | fnfvima 7239 | . . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴))) | |
19 | 5, 6, 17, 18 | mp3an12i 1461 | . 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴))) |
20 | fvex 6903 | . . . 4 ⊢ ( -us ‘𝐴) ∈ V | |
21 | 20 | snid 4661 | . . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)} |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)}) |
23 | 4, 19, 22 | ssltsepcd 27740 | 1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∀wral 3051 ∪ cun 3939 ⊆ wss 3941 {csn 4625 class class class wbr 5144 “ cima 5676 Oncon0 6365 Fn wfn 6538 ‘cfv 6543 No csur 27586 <s cslt 27587 bday cbday 27588 <<s csslt 27726 O cold 27783 L cleft 27785 R cright 27786 -us cnegs 27945 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-suc 6371 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-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-1o 8480 df-2o 8481 df-no 27589 df-slt 27590 df-bday 27591 df-sslt 27727 df-scut 27729 df-0s 27770 df-made 27787 df-old 27788 df-left 27790 df-right 27791 df-norec 27868 df-negs 27947 |
This theorem is referenced by: negsproplem7 27959 |
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