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Mirrors > Home > MPE Home > Th. List > negsproplem4 | Structured version Visualization version GIF version |
Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is simpler than 𝐵. (Contributed by Scott Fenton, 2-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) |
negsproplem4.4 | ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵)) |
Ref | Expression |
---|---|
negsproplem4 | ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negsproplem.1 | . . . . 5 ⊢ (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 | uncom 4168 | . . . . . . . 8 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴)) | |
3 | 2 | eleq2i 2831 | . . . . . . 7 ⊢ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ↔ (( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴))) |
4 | 3 | imbi1i 349 | . . . . . 6 ⊢ (((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) |
5 | 4 | 2ralbii 3126 | . . . . 5 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) |
6 | 1, 5 | sylib 218 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))))) |
7 | negsproplem4.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ No ) | |
8 | 6, 7 | negsproplem3 28077 | . . 3 ⊢ (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))) |
9 | 8 | simp3d 1143 | . 2 ⊢ (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))) |
10 | fvex 6920 | . . . 4 ⊢ ( -us ‘𝐵) ∈ V | |
11 | 10 | snid 4667 | . . 3 ⊢ ( -us ‘𝐵) ∈ {( -us ‘𝐵)} |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ {( -us ‘𝐵)}) |
13 | negsfn 28070 | . . 3 ⊢ -us Fn No | |
14 | leftssno 27934 | . . 3 ⊢ ( L ‘𝐵) ⊆ No | |
15 | negsproplem4.4 | . . . . 5 ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵)) | |
16 | bdayelon 27836 | . . . . . 6 ⊢ ( bday ‘𝐵) ∈ On | |
17 | negsproplem4.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No ) | |
18 | oldbday 27954 | . . . . . 6 ⊢ ((( bday ‘𝐵) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) | |
19 | 16, 17, 18 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵))) |
20 | 15, 19 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ( O ‘( bday ‘𝐵))) |
21 | negsproplem4.3 | . . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
22 | breq1 5151 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 <s 𝐵 ↔ 𝐴 <s 𝐵)) | |
23 | leftval 27917 | . . . . 5 ⊢ ( L ‘𝐵) = {𝑎 ∈ ( O ‘( bday ‘𝐵)) ∣ 𝑎 <s 𝐵} | |
24 | 22, 23 | elrab2 3698 | . . . 4 ⊢ (𝐴 ∈ ( L ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐴 <s 𝐵)) |
25 | 20, 21, 24 | sylanbrc 583 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵)) |
26 | fnfvima 7253 | . . 3 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝐴 ∈ ( L ‘𝐵)) → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵))) | |
27 | 13, 14, 25, 26 | mp3an12i 1464 | . 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵))) |
28 | 9, 12, 27 | ssltsepcd 27854 | 1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∪ cun 3961 ⊆ wss 3963 {csn 4631 class class class wbr 5148 “ cima 5692 Oncon0 6386 Fn wfn 6558 ‘cfv 6563 No csur 27699 <s cslt 27700 bday cbday 27701 <<s csslt 27840 O cold 27897 L cleft 27899 R cright 27900 -us cnegs 28066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sslt 27841 df-scut 27843 df-0s 27884 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-negs 28068 |
This theorem is referenced by: negsproplem7 28081 |
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