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Theorem domunsn 9167
Description: Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
domunsn (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)

Proof of Theorem domunsn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sdom0 9148 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 5147 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 327 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 4352 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 218 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 9094 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 480 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 en2sn 9081 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
109elvd 3486 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
11 endom 9019 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1210, 11syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
13 snprc 4717 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 216 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
15 vsnex 5434 . . . . . . 7 {𝑧} ∈ V
16150dom 9146 . . . . . 6 ∅ ≼ {𝑧}
1714, 16eqbrtrdi 5182 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1812, 17pm2.61i 182 . . . 4 {𝐶} ≼ {𝑧}
19 disjdifr 4473 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
20 undom 9099 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2119, 20mpan2 691 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
228, 18, 21sylancl 586 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
23 uncom 4158 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
24 simpr 484 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2524snssd 4809 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
26 undif 4482 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2725, 26sylib 218 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2823, 27eqtrid 2789 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
2922, 28breqtrd 5169 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
306, 29exlimddv 1935 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cen 8982  cdom 8983  csdm 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by:  canthp1lem1  10692
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