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Theorem ssltun1 27093
Description: Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
ssltun1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)

Proof of Theorem ssltun1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 27072 . . . 4 (𝐴 <<s 𝐶𝐴 ∈ V)
21adantr 481 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex1 27072 . . . 4 (𝐵 <<s 𝐶𝐵 ∈ V)
43adantl 482 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 ∈ V)
52, 4unexd 7680 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ∈ V)
6 ssltex2 27073 . . 3 (𝐴 <<s 𝐶𝐶 ∈ V)
76adantr 481 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 ∈ V)
8 ssltss1 27074 . . . 4 (𝐴 <<s 𝐶𝐴 No )
98adantr 481 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 No )
10 ssltss1 27074 . . . 4 (𝐵 <<s 𝐶𝐵 No )
1110adantl 482 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 No )
129, 11unssd 4144 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ⊆ No )
13 ssltss2 27075 . . 3 (𝐴 <<s 𝐶𝐶 No )
1413adantr 481 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 No )
15 elun 4106 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
16 ssltsepc 27078 . . . . . . . 8 ((𝐴 <<s 𝐶𝑥𝐴𝑦𝐶) → 𝑥 <s 𝑦)
17163exp 1119 . . . . . . 7 (𝐴 <<s 𝐶 → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
1817adantr 481 . . . . . 6 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
1918com12 32 . . . . 5 (𝑥𝐴 → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
20 ssltsepc 27078 . . . . . . . 8 ((𝐵 <<s 𝐶𝑥𝐵𝑦𝐶) → 𝑥 <s 𝑦)
21203exp 1119 . . . . . . 7 (𝐵 <<s 𝐶 → (𝑥𝐵 → (𝑦𝐶𝑥 <s 𝑦)))
2221adantl 482 . . . . . 6 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑥𝐵 → (𝑦𝐶𝑥 <s 𝑦)))
2322com12 32 . . . . 5 (𝑥𝐵 → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
2419, 23jaoi 855 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
2515, 24sylbi 216 . . 3 (𝑥 ∈ (𝐴𝐵) → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
26253imp21 1114 . 2 (((𝐴 <<s 𝐶𝐵 <<s 𝐶) ∧ 𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) → 𝑥 <s 𝑦)
275, 7, 12, 14, 26ssltd 27077 1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  wcel 2106  Vcvv 3443  cun 3906  wss 3908   class class class wbr 5103   No csur 26934   <s cslt 26935   <<s csslt 27066
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-sslt 27067
This theorem is referenced by:  scutun12  27095
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