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Theorem ssltun2 32253
Description: Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
ssltun2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))

Proof of Theorem ssltun2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 32238 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 466 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 32239 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
4 ssltex2 32239 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
5 unexg 7106 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵𝐶) ∈ V)
63, 4, 5syl2an 583 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
72, 6jca 501 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵𝐶) ∈ V))
8 ssltss1 32240 . . . 4 (𝐴 <<s 𝐵𝐴 No )
98adantr 466 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 32241 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
1110adantr 466 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 32241 . . . . 5 (𝐴 <<s 𝐶𝐶 No )
1312adantl 467 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 3940 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 ssltsep 32242 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1615adantr 466 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
17 ssltsep 32242 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1817adantl 467 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 3945 . . . . . 6 (∀𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
2019ralbii 3129 . . . . 5 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
21 r19.26 3212 . . . . 5 (∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦) ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2220, 21bitri 264 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2316, 18, 22sylanbrc 572 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)
249, 14, 233jca 1122 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦))
25 brsslt 32237 . 2 (𝐴 <<s (𝐵𝐶) ↔ ((𝐴 ∈ V ∧ (𝐵𝐶) ∈ V) ∧ (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)))
267, 24, 25sylanbrc 572 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wcel 2145  wral 3061  Vcvv 3351  cun 3721  wss 3723   class class class wbr 4786   No csur 32130   <s cslt 32131   <<s csslt 32233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-sslt 32234
This theorem is referenced by:  scutun12  32254
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