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Theorem ssltun2 33344
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 33329 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 484 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 33330 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
4 ssltex2 33330 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
5 unexg 7457 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵𝐶) ∈ V)
63, 4, 5syl2an 598 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
72, 6jca 515 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵𝐶) ∈ V))
8 ssltss1 33331 . . . 4 (𝐴 <<s 𝐵𝐴 No )
98adantr 484 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 33332 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
1110adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 33332 . . . . 5 (𝐴 <<s 𝐶𝐶 No )
1312adantl 485 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4137 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 ssltsep 33333 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1615adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
17 ssltsep 33333 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1817adantl 485 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 4142 . . . . . 6 (∀𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
2019ralbii 3157 . . . . 5 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
21 r19.26 3162 . . . . 5 (∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦) ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2220, 21bitri 278 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2316, 18, 22sylanbrc 586 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)
249, 14, 233jca 1125 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦))
25 brsslt 33328 . 2 (𝐴 <<s (𝐵𝐶) ↔ ((𝐴 ∈ V ∧ (𝐵𝐶) ∈ V) ∧ (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)))
267, 24, 25sylanbrc 586 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2114  wral 3130  Vcvv 3469  cun 3906  wss 3908   class class class wbr 5042   No csur 33221   <s cslt 33222   <<s csslt 33324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-sslt 33325
This theorem is referenced by:  scutun12  33345
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