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Theorem snnen2o 8691
Description: A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.)
Assertion
Ref Expression
snnen2o ¬ {𝐴} ≈ 2o

Proof of Theorem snnen2o
StepHypRef Expression
1 1onn 8248 . . . 4 1o ∈ ω
2 php5 8689 . . . 4 (1o ∈ ω → ¬ 1o ≈ suc 1o)
31, 2ax-mp 5 . . 3 ¬ 1o ≈ suc 1o
4 ensn1g 8557 . . 3 (𝐴 ∈ V → {𝐴} ≈ 1o)
5 df-2o 8086 . . . . . 6 2o = suc 1o
65eqcomi 2833 . . . . 5 suc 1o = 2o
76breq2i 5055 . . . 4 (1o ≈ suc 1o ↔ 1o ≈ 2o)
8 ensymb 8540 . . . . . 6 ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴})
9 entr 8544 . . . . . . 7 ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o)
109ex 416 . . . . . 6 (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o))
118, 10sylbi 220 . . . . 5 ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o))
1211con3rr3 158 . . . 4 (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
137, 12sylnbi 333 . . 3 (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
143, 4, 13mpsyl 68 . 2 (𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
15 2on0 8096 . . . 4 2o ≠ ∅
16 ensymb 8540 . . . . 5 (∅ ≈ 2o ↔ 2o ≈ ∅)
17 en0 8555 . . . . 5 (2o ≈ ∅ ↔ 2o = ∅)
1816, 17bitri 278 . . . 4 (∅ ≈ 2o ↔ 2o = ∅)
1915, 18nemtbir 3108 . . 3 ¬ ∅ ≈ 2o
20 snprc 4634 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
2120biimpi 219 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
2221breq1d 5057 . . 3 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o))
2319, 22mtbiri 330 . 2 𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
2414, 23pm2.61i 185 1 ¬ {𝐴} ≈ 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  Vcvv 3479  c0 4274  {csn 4548   class class class wbr 5047  suc csuc 6174  ωcom 7563  1oc1o 8078  2oc2o 8079  cen 8489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495
This theorem is referenced by:  pmtrsn  18636  trivnsimpgd  19208
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