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Theorem ssltun2 32244
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 32229 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 472 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 32230 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
4 ssltex2 32230 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
5 unexg 7126 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵𝐶) ∈ V)
63, 4, 5syl2an 495 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
72, 6jca 555 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵𝐶) ∈ V))
8 ssltss1 32231 . . . 4 (𝐴 <<s 𝐵𝐴 No )
98adantr 472 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 32232 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
1110adantr 472 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 32232 . . . . 5 (𝐴 <<s 𝐶𝐶 No )
1312adantl 473 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 3933 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 ssltsep 32233 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1615adantr 472 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
17 ssltsep 32233 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1817adantl 473 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 3938 . . . . . 6 (∀𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
2019ralbii 3119 . . . . 5 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
21 r19.26 3203 . . . . 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 701 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)
249, 14, 233jca 1123 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦))
25 brsslt 32228 . 2 (𝐴 <<s (𝐵𝐶) ↔ ((𝐴 ∈ V ∧ (𝐵𝐶) ∈ V) ∧ (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)))
267, 24, 25sylanbrc 701 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2140  wral 3051  Vcvv 3341  cun 3714  wss 3716   class class class wbr 4805   No csur 32121   <s cslt 32122   <<s csslt 32224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-xp 5273  df-sslt 32225
This theorem is referenced by:  scutun12  32245
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