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Theorem negsproplem7 28081
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 27836 . . . 4 ( bday 𝐴) ∈ On
21onordi 6497 . . 3 Ord ( bday 𝐴)
3 bdayelon 27836 . . . 4 ( bday 𝐵) ∈ On
43onordi 6497 . . 3 Ord ( bday 𝐵)
5 ordtri3or 6418 . . 3 ((Ord ( bday 𝐴) ∧ Ord ( bday 𝐵)) → (( bday 𝐴) ∈ ( bday 𝐵) ∨ ( bday 𝐴) = ( bday 𝐵) ∨ ( bday 𝐵) ∈ ( bday 𝐴)))
62, 4, 5mp2an 692 . 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 28078 . . . 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 28080 . . . 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 28079 . . . 4 ((𝜑 ∧ ( bday 𝐵) ∈ ( bday 𝐴)) → ( -us𝐵) <s ( -us𝐴))
3130ex 412 . . 3 (𝜑 → (( bday 𝐵) ∈ ( bday 𝐴) → ( -us𝐵) <s ( -us𝐴)))
3217, 24, 313jaod 1428 . 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 1085   = wceq 1537  wcel 2106  wral 3059  cun 3961   class class class wbr 5148  Ord word 6385  cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701   -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:  negsprop  28082
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