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Theorem ssltun1 33377
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 33363 . . . 4 (𝐴 <<s 𝐶𝐴 ∈ V)
2 ssltex1 33363 . . . 4 (𝐵 <<s 𝐶𝐵 ∈ V)
3 unexg 7456 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3syl2an 598 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ∈ V)
5 ssltex2 33364 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantr 484 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 ∈ V)
74, 6jca 515 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V))
8 ssltss1 33365 . . . . 5 (𝐴 <<s 𝐶𝐴 No )
98adantr 484 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 No )
10 ssltss1 33365 . . . . 5 (𝐵 <<s 𝐶𝐵 No )
1110adantl 485 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 No )
129, 11unssd 4116 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ⊆ No )
13 ssltss2 33366 . . . 4 (𝐵 <<s 𝐶𝐶 No )
1413adantl 485 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 No )
15 ssltsep 33367 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1615adantr 484 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
17 ssltsep 33367 . . . . 5 (𝐵 <<s 𝐶 → ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦)
1817adantl 485 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 4121 . . . 4 (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦 ∧ ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦))
2016, 18, 19sylanbrc 586 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦)
2112, 14, 203jca 1125 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ((𝐴𝐵) ⊆ No 𝐶 No ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦))
22 brsslt 33362 . 2 ((𝐴𝐵) <<s 𝐶 ↔ (((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V) ∧ ((𝐴𝐵) ⊆ No 𝐶 No ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦)))
237, 21, 22sylanbrc 586 1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2112  wral 3109  Vcvv 3444  cun 3882  wss 3884   class class class wbr 5033   No csur 33255   <s cslt 33256   <<s csslt 33358
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-sslt 33359
This theorem is referenced by:  scutun12  33379
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