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Theorem negsproplem7 27920
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 27683 . . . 4 ( bday 𝐴) ∈ On
21onordi 6474 . . 3 Ord ( bday 𝐴)
3 bdayelon 27683 . . . 4 ( bday 𝐵) ∈ On
43onordi 6474 . . 3 Ord ( bday 𝐵)
5 ordtri3or 6395 . . 3 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → (( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐴) = ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)))
62, 4, 5mp2an 691 . 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 480 . . . . 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 480 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → 𝐴 No )
11 negsproplem4.2 . . . . . 6 (𝜑𝐵 No )
1211adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → 𝐵 No )
13 negsproplem4.3 . . . . . 6 (𝜑𝐴 <s 𝐵)
1413adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → 𝐴 <s 𝐵)
15 simpr 484 . . . . 5 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → ( bday 𝐴) ∈ ( bday 𝐵))
168, 10, 12, 14, 15negsproplem4 27917 . . . 4 ((𝜑 ∧ ( bday 𝐴) ∈ ( bday 𝐵)) → ( -us𝐵) <s ( -us𝐴))
1716ex 412 . . 3 (𝜑 → (( bday 𝐴) ∈ ( bday 𝐵) → ( -us𝐵) <s ( -us𝐴)))
187adantr 480 . . . . 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 480 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → 𝐴 No )
2011adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → 𝐵 No )
2113adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → 𝐴 <s 𝐵)
22 simpr 484 . . . . 5 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → ( bday 𝐴) = ( bday 𝐵))
2318, 19, 20, 21, 22negsproplem6 27919 . . . 4 ((𝜑 ∧ ( bday 𝐴) = ( bday 𝐵)) → ( -us𝐵) <s ( -us𝐴))
2423ex 412 . . 3 (𝜑 → (( bday 𝐴) = ( bday 𝐵) → ( -us𝐵) <s ( -us𝐴)))
257adantr 480 . . . . 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 480 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → 𝐴 No )
2711adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → 𝐵 No )
2813adantr 480 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → 𝐴 <s 𝐵)
29 simpr 484 . . . . 5 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → ( bday 𝐵) ∈ ( bday 𝐴))
3025, 26, 27, 28, 29negsproplem5 27918 . . . 4 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → ( -us𝐵) <s ( -us𝐴))
3130ex 412 . . 3 (𝜑 → (( bday 𝐵) ∈ ( bday 𝐴) → ( -us𝐵) <s ( -us𝐴)))
3217, 24, 313jaod 1426 . 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 395  w3o 1084   = wceq 1534  wcel 2099  wral 3056  cun 3942   class class class wbr 5142  Ord word 6362  cfv 6542   No csur 27547   <s cslt 27548   bday cbday 27549   -us cnegs 27906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-suc 6369  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-1o 8478  df-2o 8479  df-no 27550  df-slt 27551  df-bday 27552  df-sslt 27688  df-scut 27690  df-0s 27731  df-made 27748  df-old 27749  df-left 27751  df-right 27752  df-norec 27829  df-negs 27908
This theorem is referenced by:  negsprop  27921
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