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Theorem unsnen 9969
Description: Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
Hypotheses
Ref Expression
unsnen.1 𝐴 ∈ V
unsnen.2 𝐵 ∈ V
Assertion
Ref Expression
unsnen 𝐵𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴))

Proof of Theorem unsnen
StepHypRef Expression
1 disjsn 4646 . . 3 ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝐴)
2 cardon 9367 . . . . . 6 (card‘𝐴) ∈ On
32onordi 6294 . . . . 5 Ord (card‘𝐴)
4 orddisj 6228 . . . . 5 (Ord (card‘𝐴) → ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)
53, 4ax-mp 5 . . . 4 ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅
6 unsnen.1 . . . . . . 7 𝐴 ∈ V
76cardid 9963 . . . . . 6 (card‘𝐴) ≈ 𝐴
87ensymi 8553 . . . . 5 𝐴 ≈ (card‘𝐴)
9 unsnen.2 . . . . . 6 𝐵 ∈ V
10 fvex 6682 . . . . . 6 (card‘𝐴) ∈ V
11 en2sn 8587 . . . . . 6 ((𝐵 ∈ V ∧ (card‘𝐴) ∈ V) → {𝐵} ≈ {(card‘𝐴)})
129, 10, 11mp2an 688 . . . . 5 {𝐵} ≈ {(card‘𝐴)}
13 unen 8590 . . . . 5 (((𝐴 ≈ (card‘𝐴) ∧ {𝐵} ≈ {(card‘𝐴)}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)}))
148, 12, 13mpanl12 698 . . . 4 (((𝐴 ∩ {𝐵}) = ∅ ∧ ((card‘𝐴) ∩ {(card‘𝐴)}) = ∅) → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)}))
155, 14mpan2 687 . . 3 ((𝐴 ∩ {𝐵}) = ∅ → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)}))
161, 15sylbir 236 . 2 𝐵𝐴 → (𝐴 ∪ {𝐵}) ≈ ((card‘𝐴) ∪ {(card‘𝐴)}))
17 df-suc 6196 . 2 suc (card‘𝐴) = ((card‘𝐴) ∪ {(card‘𝐴)})
1816, 17breqtrrdi 5105 1 𝐵𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cun 3938  cin 3939  c0 4295  {csn 4564   class class class wbr 5063  Ord word 6189  suc csuc 6192  cfv 6354  cen 8500  cardccrd 9358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-ac2 9879
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-wrecs 7943  df-recs 8004  df-1o 8098  df-er 8284  df-en 8504  df-card 9362  df-ac 9536
This theorem is referenced by: (None)
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