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Theorem negsproplem1 28075
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 6907 . . . . . 6 (𝑥 = 𝑋 → ( bday 𝑥) = ( bday 𝑋))
76uneq1d 4177 . . . . 5 (𝑥 = 𝑋 → (( bday 𝑥) ∪ ( bday 𝑦)) = (( bday 𝑋) ∪ ( bday 𝑦)))
87eleq1d 2824 . . . 4 (𝑥 = 𝑋 → ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) ↔ (( bday 𝑋) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵))))
9 fveq2 6907 . . . . . 6 (𝑥 = 𝑋 → ( -us𝑥) = ( -us𝑋))
109eleq1d 2824 . . . . 5 (𝑥 = 𝑋 → (( -us𝑥) ∈ No ↔ ( -us𝑋) ∈ No ))
11 breq1 5151 . . . . . 6 (𝑥 = 𝑋 → (𝑥 <s 𝑦𝑋 <s 𝑦))
129breq2d 5160 . . . . . 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 6907 . . . . . 6 (𝑦 = 𝑌 → ( bday 𝑦) = ( bday 𝑌))
1716uneq2d 4178 . . . . 5 (𝑦 = 𝑌 → (( bday 𝑋) ∪ ( bday 𝑦)) = (( bday 𝑋) ∪ ( bday 𝑌)))
1817eleq1d 2824 . . . 4 (𝑦 = 𝑌 → ((( bday 𝑋) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) ↔ (( bday 𝑋) ∪ ( bday 𝑌)) ∈ (( bday 𝐴) ∪ ( bday 𝐵))))
19 breq2 5152 . . . . . 6 (𝑦 = 𝑌 → (𝑋 <s 𝑦𝑋 <s 𝑌))
20 fveq2 6907 . . . . . . 7 (𝑦 = 𝑌 → ( -us𝑦) = ( -us𝑌))
2120breq1d 5158 . . . . . 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 3633 . 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 1537  wcel 2106  wral 3059  cun 3961   class class class wbr 5148  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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  negsproplem2  28076  negsproplem6  28080
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