MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cda1en Structured version   Visualization version   GIF version

Theorem cda1en 8858
Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cda1en ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 +𝑐 1𝑜) ≈ suc 𝐴)

Proof of Theorem cda1en
StepHypRef Expression
1 enrefg 7851 . . . 4 (𝐴𝑉𝐴𝐴)
21adantr 480 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → 𝐴𝐴)
3 ensn1g 7885 . . . . 5 (𝐴𝑉 → {𝐴} ≈ 1𝑜)
43ensymd 7871 . . . 4 (𝐴𝑉 → 1𝑜 ≈ {𝐴})
54adantr 480 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → 1𝑜 ≈ {𝐴})
6 simpr 476 . . . 4 ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → ¬ 𝐴𝐴)
7 disjsn 4192 . . . 4 ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐴)
86, 7sylibr 223 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 ∩ {𝐴}) = ∅)
9 cdaenun 8857 . . 3 ((𝐴𝐴 ∧ 1𝑜 ≈ {𝐴} ∧ (𝐴 ∩ {𝐴}) = ∅) → (𝐴 +𝑐 1𝑜) ≈ (𝐴 ∪ {𝐴}))
102, 5, 8, 9syl3anc 1318 . 2 ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 +𝑐 1𝑜) ≈ (𝐴 ∪ {𝐴}))
11 df-suc 5632 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
1210, 11syl6breqr 4620 1 ((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 +𝑐 1𝑜) ≈ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  cun 3538  cin 3539  c0 3874  {csn 4125   class class class wbr 4578  suc csuc 5628  (class class class)co 6527  1𝑜c1o 7418  cen 7816   +𝑐 ccda 8850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1o 7425  df-er 7607  df-en 7820  df-cda 8851
This theorem is referenced by:  pm110.643ALT  8861  pwsdompw  8887
  Copyright terms: Public domain W3C validator