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Theorem negsproplem1 28061
Description: Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-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)))))
negsproplem1.1 (𝜑𝑋 No )
negsproplem1.2 (𝜑𝑌 No )
negsproplem1.3 (𝜑 → (( bday 𝑋) ∪ ( bday 𝑌)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
Assertion
Ref Expression
negsproplem1 (𝜑 → (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us𝑌) <s ( -us𝑋))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑦,𝑌
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem negsproplem1
StepHypRef Expression
1 negsproplem1.1 . . 3 (𝜑𝑋 No )
2 negsproplem1.2 . . 3 (𝜑𝑌 No )
31, 2jca 511 . 2 (𝜑 → (𝑋 No 𝑌 No ))
4 negsproplem.1 . 2 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
5 negsproplem1.3 . 2 (𝜑 → (( bday 𝑋) ∪ ( bday 𝑌)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
6 fveq2 6905 . . . . . 6 (𝑥 = 𝑋 → ( bday 𝑥) = ( bday 𝑋))
76uneq1d 4166 . . . . 5 (𝑥 = 𝑋 → (( bday 𝑥) ∪ ( bday 𝑦)) = (( bday 𝑋) ∪ ( bday 𝑦)))
87eleq1d 2825 . . . 4 (𝑥 = 𝑋 → ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) ↔ (( bday 𝑋) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵))))
9 fveq2 6905 . . . . . 6 (𝑥 = 𝑋 → ( -us𝑥) = ( -us𝑋))
109eleq1d 2825 . . . . 5 (𝑥 = 𝑋 → (( -us𝑥) ∈ No ↔ ( -us𝑋) ∈ No ))
11 breq1 5145 . . . . . 6 (𝑥 = 𝑋 → (𝑥 <s 𝑦𝑋 <s 𝑦))
129breq2d 5154 . . . . . 6 (𝑥 = 𝑋 → (( -us𝑦) <s ( -us𝑥) ↔ ( -us𝑦) <s ( -us𝑋)))
1311, 12imbi12d 344 . . . . 5 (𝑥 = 𝑋 → ((𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)) ↔ (𝑋 <s 𝑦 → ( -us𝑦) <s ( -us𝑋))))
1410, 13anbi12d 632 . . . 4 (𝑥 = 𝑋 → ((( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥))) ↔ (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us𝑦) <s ( -us𝑋)))))
158, 14imbi12d 344 . . 3 (𝑥 = 𝑋 → (((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))) ↔ ((( bday 𝑋) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us𝑦) <s ( -us𝑋))))))
16 fveq2 6905 . . . . . 6 (𝑦 = 𝑌 → ( bday 𝑦) = ( bday 𝑌))
1716uneq2d 4167 . . . . 5 (𝑦 = 𝑌 → (( bday 𝑋) ∪ ( bday 𝑦)) = (( bday 𝑋) ∪ ( bday 𝑌)))
1817eleq1d 2825 . . . 4 (𝑦 = 𝑌 → ((( bday 𝑋) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) ↔ (( bday 𝑋) ∪ ( bday 𝑌)) ∈ (( bday 𝐴) ∪ ( bday 𝐵))))
19 breq2 5146 . . . . . 6 (𝑦 = 𝑌 → (𝑋 <s 𝑦𝑋 <s 𝑌))
20 fveq2 6905 . . . . . . 7 (𝑦 = 𝑌 → ( -us𝑦) = ( -us𝑌))
2120breq1d 5152 . . . . . 6 (𝑦 = 𝑌 → (( -us𝑦) <s ( -us𝑋) ↔ ( -us𝑌) <s ( -us𝑋)))
2219, 21imbi12d 344 . . . . 5 (𝑦 = 𝑌 → ((𝑋 <s 𝑦 → ( -us𝑦) <s ( -us𝑋)) ↔ (𝑋 <s 𝑌 → ( -us𝑌) <s ( -us𝑋))))
2322anbi2d 630 . . . 4 (𝑦 = 𝑌 → ((( -us𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us𝑦) <s ( -us𝑋))) ↔ (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us𝑌) <s ( -us𝑋)))))
2418, 23imbi12d 344 . . 3 (𝑦 = 𝑌 → (((( bday 𝑋) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us𝑦) <s ( -us𝑋)))) ↔ ((( bday 𝑋) ∪ ( bday 𝑌)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us𝑌) <s ( -us𝑋))))))
2515, 24rspc2v 3632 . 2 ((𝑋 No 𝑌 No ) → (∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))) → ((( bday 𝑋) ∪ ( bday 𝑌)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us𝑌) <s ( -us𝑋))))))
263, 4, 5, 25syl3c 66 1 (𝜑 → (( -us𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us𝑌) <s ( -us𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  cun 3948   class class class wbr 5142  cfv 6560   No csur 27685   <s cslt 27686   bday cbday 27687   -us cnegs 28052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568
This theorem is referenced by:  negsproplem2  28062  negsproplem6  28066
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