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Theorem domunsn 9129
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 9110 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 5152 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 326 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 4345 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 217 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 9056 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 481 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 en2sn 9043 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
109elvd 3481 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
11 endom 8977 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1210, 11syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
13 snprc 4721 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 215 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
15 vsnex 5429 . . . . . . 7 {𝑧} ∈ V
16150dom 9108 . . . . . 6 ∅ ≼ {𝑧}
1714, 16eqbrtrdi 5187 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1812, 17pm2.61i 182 . . . 4 {𝐶} ≼ {𝑧}
19 disjdifr 4472 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
20 undom 9061 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2119, 20mpan2 689 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
228, 18, 21sylancl 586 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
23 uncom 4153 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
24 simpr 485 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2524snssd 4812 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
26 undif 4481 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2725, 26sylib 217 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2823, 27eqtrid 2784 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
2922, 28breqtrd 5174 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
306, 29exlimddv 1938 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628   class class class wbr 5148  cen 8938  cdom 8939  csdm 8940
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-en 8942  df-dom 8943  df-sdom 8944
This theorem is referenced by:  canthp1lem1  10649
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