MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  negsproplem7 Structured version   Visualization version   GIF version

Theorem negsproplem7 27962
Description: Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 3-Feb-2025.)
Hypotheses
Ref Expression
negsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
negsproplem4.1 (𝜑𝐴 No )
negsproplem4.2 (𝜑𝐵 No )
negsproplem4.3 (𝜑𝐴 <s 𝐵)
Assertion
Ref Expression
negsproplem7 (𝜑 → ( -us𝐵) <s ( -us𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem7
StepHypRef Expression
1 bdayelon 27725 . . . 4 ( bday 𝐴) ∈ On
21onordi 6475 . . 3 Ord ( bday 𝐴)
3 bdayelon 27725 . . . 4 ( bday 𝐵) ∈ On
43onordi 6475 . . 3 Ord ( bday 𝐵)
5 ordtri3or 6396 . . 3 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → (( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐴) = ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)))
62, 4, 5mp2an 690 . 2 (( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐴) = ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴))
7 negsproplem.1 . . . . . 6 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
87adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
9 negsproplem4.1 . . . . . 6 (𝜑𝐴 No )
109adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → 𝐴 No )
11 negsproplem4.2 . . . . . 6 (𝜑𝐵 No )
1211adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → 𝐵 No )
13 negsproplem4.3 . . . . . 6 (𝜑𝐴 <s 𝐵)
1413adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → 𝐴 <s 𝐵)
15 simpr 483 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → ( bday 𝐴) ∈ ( bday 𝐵))
168, 10, 12, 14, 15negsproplem4 27959 . . . 4 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → ( -us𝐵) <s ( -us𝐴))
1716ex 411 . . 3 (𝜑 → (( bday 𝐴) ∈ ( bday 𝐵) → ( -us𝐵) <s ( -us𝐴)))
187adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
199adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → 𝐴 No )
2011adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → 𝐵 No )
2113adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → 𝐴 <s 𝐵)
22 simpr 483 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → ( bday 𝐴) = ( bday 𝐵))
2318, 19, 20, 21, 22negsproplem6 27961 . . . 4 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → ( -us𝐵) <s ( -us𝐴))
2423ex 411 . . 3 (𝜑 → (( bday 𝐴) = ( bday 𝐵) → ( -us𝐵) <s ( -us𝐴)))
257adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
269adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → 𝐴 No )
2711adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → 𝐵 No )
2813adantr 479 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → 𝐴 <s 𝐵)
29 simpr 483 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → ( bday 𝐵) ∈ ( bday 𝐴))
3025, 26, 27, 28, 29negsproplem5 27960 . . . 4 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → ( -us𝐵) <s ( -us𝐴))
3130ex 411 . . 3 (𝜑 → (( bday 𝐵) ∈ ( bday 𝐴) → ( -us𝐵) <s ( -us𝐴)))
3217, 24, 313jaod 1425 . 2 (𝜑 → ((( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐴) = ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)) → ( -us𝐵) <s ( -us𝐴)))
336, 32mpi 20 1 (𝜑 → ( -us𝐵) <s ( -us𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3o 1083   = wceq 1533  wcel 2098  wral 3051  cun 3938   class class class wbr 5143  Ord word 6363  cfv 6542   No csur 27589   <s cslt 27590   bday cbday 27591   -us cnegs 27948
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-1o 8483  df-2o 8484  df-no 27592  df-slt 27593  df-bday 27594  df-sslt 27730  df-scut 27732  df-0s 27773  df-made 27790  df-old 27791  df-left 27793  df-right 27794  df-norec 27871  df-negs 27950
This theorem is referenced by:  negsprop  27963
  Copyright terms: Public domain W3C validator