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Theorem ssltun1 27854
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 27832 . . . 4 (𝐴 <<s 𝐶𝐴 ∈ V)
21adantr 480 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex1 27832 . . . 4 (𝐵 <<s 𝐶𝐵 ∈ V)
43adantl 481 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 ∈ V)
52, 4unexd 7775 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ∈ V)
6 ssltex2 27833 . . 3 (𝐴 <<s 𝐶𝐶 ∈ V)
76adantr 480 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 ∈ V)
8 ssltss1 27834 . . . 4 (𝐴 <<s 𝐶𝐴 No )
98adantr 480 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 No )
10 ssltss1 27834 . . . 4 (𝐵 <<s 𝐶𝐵 No )
1110adantl 481 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 No )
129, 11unssd 4191 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ⊆ No )
13 ssltss2 27835 . . 3 (𝐴 <<s 𝐶𝐶 No )
1413adantr 480 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 No )
15 elun 4152 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
16 ssltsepc 27839 . . . . . . . 8 ((𝐴 <<s 𝐶𝑥𝐴𝑦𝐶) → 𝑥 <s 𝑦)
17163exp 1119 . . . . . . 7 (𝐴 <<s 𝐶 → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
1817adantr 480 . . . . . 6 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
1918com12 32 . . . . 5 (𝑥𝐴 → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
20 ssltsepc 27839 . . . . . . . 8 ((𝐵 <<s 𝐶𝑥𝐵𝑦𝐶) → 𝑥 <s 𝑦)
21203exp 1119 . . . . . . 7 (𝐵 <<s 𝐶 → (𝑥𝐵 → (𝑦𝐶𝑥 <s 𝑦)))
2221adantl 481 . . . . . 6 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑥𝐵 → (𝑦𝐶𝑥 <s 𝑦)))
2322com12 32 . . . . 5 (𝑥𝐵 → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
2419, 23jaoi 857 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
2515, 24sylbi 217 . . 3 (𝑥 ∈ (𝐴𝐵) → ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝑦𝐶𝑥 <s 𝑦)))
26253imp21 1113 . 2 (((𝐴 <<s 𝐶𝐵 <<s 𝐶) ∧ 𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) → 𝑥 <s 𝑦)
275, 7, 12, 14, 26ssltd 27837 1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wcel 2107  Vcvv 3479  cun 3948  wss 3950   class class class wbr 5142   No csur 27685   <s cslt 27686   <<s csslt 27826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-sslt 27827
This theorem is referenced by:  scutun12  27856
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