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Theorem domunsn 8266
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 8248 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 4790 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 316 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 136 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 4077 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 208 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 8199 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 466 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 vex 3354 . . . . . . 7 𝑧 ∈ V
10 en2sn 8193 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
119, 10mpan2 671 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
12 endom 8136 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1311, 12syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
14 snprc 4389 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1514biimpi 206 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
16 snex 5036 . . . . . . 7 {𝑧} ∈ V
17160dom 8246 . . . . . 6 ∅ ≼ {𝑧}
1815, 17syl6eqbr 4825 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1913, 18pm2.61i 176 . . . 4 {𝐶} ≼ {𝑧}
20 incom 3956 . . . . . 6 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝐵 ∖ {𝑧}))
21 disjdif 4182 . . . . . 6 ({𝑧} ∩ (𝐵 ∖ {𝑧})) = ∅
2220, 21eqtri 2793 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
23 undom 8204 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2422, 23mpan2 671 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
258, 19, 24sylancl 574 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
26 uncom 3908 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
27 simpr 471 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2827snssd 4475 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
29 undif 4191 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3028, 29sylib 208 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3126, 30syl5eq 2817 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
3225, 31breqtrd 4812 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
336, 32exlimddv 2015 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  {csn 4316   class class class wbr 4786  cen 8106  cdom 8107  csdm 8108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-suc 5872  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-1o 7713  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112
This theorem is referenced by:  canthp1lem1  9676
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