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Theorem domunsn 9127
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 9108 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 5153 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 327 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 4346 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 217 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 9054 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 482 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 en2sn 9041 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
109elvd 3482 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
11 endom 8975 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1210, 11syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
13 snprc 4722 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 215 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
15 vsnex 5430 . . . . . . 7 {𝑧} ∈ V
16150dom 9106 . . . . . 6 ∅ ≼ {𝑧}
1714, 16eqbrtrdi 5188 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1812, 17pm2.61i 182 . . . 4 {𝐶} ≼ {𝑧}
19 disjdifr 4473 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
20 undom 9059 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2119, 20mpan2 690 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
228, 18, 21sylancl 587 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
23 uncom 4154 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
24 simpr 486 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2524snssd 4813 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
26 undif 4482 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2725, 26sylib 217 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2823, 27eqtrid 2785 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
2922, 28breqtrd 5175 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
306, 29exlimddv 1939 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  {csn 4629   class class class wbr 5149  cen 8936  cdom 8937  csdm 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-en 8940  df-dom 8941  df-sdom 8942
This theorem is referenced by:  canthp1lem1  10647
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