Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssltun1 Structured version   Visualization version   GIF version

Theorem ssltun1 32241
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 32227 . . . 4 (𝐴 <<s 𝐶𝐴 ∈ V)
2 ssltex1 32227 . . . 4 (𝐵 <<s 𝐶𝐵 ∈ V)
3 unexg 7192 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3syl2an 585 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ∈ V)
5 ssltex2 32228 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantr 468 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 ∈ V)
74, 6jca 503 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V))
8 ssltss1 32229 . . . . 5 (𝐴 <<s 𝐶𝐴 No )
98adantr 468 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 No )
10 ssltss1 32229 . . . . 5 (𝐵 <<s 𝐶𝐵 No )
1110adantl 469 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 No )
129, 11unssd 3995 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ⊆ No )
13 ssltss2 32230 . . . 4 (𝐵 <<s 𝐶𝐶 No )
1413adantl 469 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 No )
15 ssltsep 32231 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1615adantr 468 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
17 ssltsep 32231 . . . . 5 (𝐵 <<s 𝐶 → ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦)
1817adantl 469 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 4000 . . . 4 (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦 ∧ ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦))
2016, 18, 19sylanbrc 574 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦)
2112, 14, 203jca 1151 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ((𝐴𝐵) ⊆ No 𝐶 No ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦))
22 brsslt 32226 . 2 ((𝐴𝐵) <<s 𝐶 ↔ (((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V) ∧ ((𝐴𝐵) ⊆ No 𝐶 No ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦)))
237, 21, 22sylanbrc 574 1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wcel 2157  wral 3103  Vcvv 3398  cun 3774  wss 3776   class class class wbr 4851   No csur 32119   <s cslt 32120   <<s csslt 32222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-xp 5324  df-sslt 32223
This theorem is referenced by:  scutun12  32243
  Copyright terms: Public domain W3C validator