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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 34316 | . . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)))) |
4 | 3 | simp2d 1143 | . 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)}) |
5 | negsfn 34310 | . . 3 ⊢ -us Fn No | |
6 | rightssno 27161 | . . 3 ⊢ ( R ‘𝐴) ⊆ No | |
7 | negsproplem5.4 | . . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) | |
8 | bdayelon 27067 | . . . . . 6 ⊢ ( bday ‘𝐴) ∈ On | |
9 | negsproplem4.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No ) | |
10 | oldbday 27178 | . . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴))) | |
11 | 8, 9, 10 | sylancr 587 | . . . . 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 5107 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴 <s 𝑏 ↔ 𝐴 <s 𝐵)) | |
15 | rightval 27145 | . . . . 5 ⊢ ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑏} | |
16 | 14, 15 | elrab2 3646 | . . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵)) |
17 | 12, 13, 16 | sylanbrc 583 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴)) |
18 | fnfvima 7179 | . . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴))) | |
19 | 5, 6, 17, 18 | mp3an12i 1465 | . 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴))) |
20 | fvex 6852 | . . . 4 ⊢ ( -us ‘𝐴) ∈ V | |
21 | 20 | snid 4620 | . . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)} |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)}) |
23 | 4, 19, 22 | ssltsepcd 27084 | 1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3062 ∪ cun 3906 ⊆ wss 3908 {csn 4584 class class class wbr 5103 “ cima 5634 Oncon0 6315 Fn wfn 6488 ‘cfv 6493 No csur 26939 <s cslt 26940 bday cbday 26941 <<s csslt 27071 O cold 27124 L cleft 27126 R cright 27127 -us cnegs 34306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-1o 8404 df-2o 8405 df-no 26942 df-slt 26943 df-bday 26944 df-sslt 27072 df-scut 27074 df-0s 27114 df-made 27128 df-old 27129 df-left 27131 df-right 27132 df-norec 34246 df-negs 34308 |
This theorem is referenced by: negsproplem7 34320 |
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