MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domunsn Structured version   Visualization version   GIF version

Theorem domunsn 8095
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 8077 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 4648 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 317 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 134 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 3922 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 208 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 8028 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 481 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 vex 3198 . . . . . . 7 𝑧 ∈ V
10 en2sn 8022 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
119, 10mpan2 706 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
12 endom 7967 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1311, 12syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
14 snprc 4244 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1514biimpi 206 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
16 snex 4899 . . . . . . 7 {𝑧} ∈ V
17160dom 8075 . . . . . 6 ∅ ≼ {𝑧}
1815, 17syl6eqbr 4683 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1913, 18pm2.61i 176 . . . 4 {𝐶} ≼ {𝑧}
20 incom 3797 . . . . . 6 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝐵 ∖ {𝑧}))
21 disjdif 4031 . . . . . 6 ({𝑧} ∩ (𝐵 ∖ {𝑧})) = ∅
2220, 21eqtri 2642 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
23 undom 8033 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2422, 23mpan2 706 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
258, 19, 24sylancl 693 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
26 uncom 3749 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
27 simpr 477 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2827snssd 4331 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
29 undif 4040 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3028, 29sylib 208 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3126, 30syl5eq 2666 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
3225, 31breqtrd 4670 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
336, 32exlimddv 1861 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {csn 4168   class class class wbr 4644  cen 7937  cdom 7938  csdm 7939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943
This theorem is referenced by:  canthp1lem1  9459
  Copyright terms: Public domain W3C validator