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Theorem snnen2o 8439
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 8005 . . . 4 1o ∈ ω
2 php5 8438 . . . 4 (1o ∈ ω → ¬ 1o ≈ suc 1o)
31, 2ax-mp 5 . . 3 ¬ 1o ≈ suc 1o
4 ensn1g 8308 . . 3 (𝐴 ∈ V → {𝐴} ≈ 1o)
5 df-2o 7846 . . . . . 6 2o = suc 1o
65eqcomi 2787 . . . . 5 suc 1o = 2o
76breq2i 4896 . . . 4 (1o ≈ suc 1o ↔ 1o ≈ 2o)
8 ensymb 8291 . . . . . 6 ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴})
9 entr 8295 . . . . . . 7 ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o)
109ex 403 . . . . . 6 (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o))
118, 10sylbi 209 . . . . 5 ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o))
1211con3rr3 153 . . . 4 (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
137, 12sylnbi 322 . . 3 (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
143, 4, 13mpsyl 68 . 2 (𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
15 2on0 7855 . . . 4 2o ≠ ∅
16 ensymb 8291 . . . . 5 (∅ ≈ 2o ↔ 2o ≈ ∅)
17 en0 8306 . . . . 5 (2o ≈ ∅ ↔ 2o = ∅)
1816, 17bitri 267 . . . 4 (∅ ≈ 2o ↔ 2o = ∅)
1915, 18nemtbir 3065 . . 3 ¬ ∅ ≈ 2o
20 snprc 4484 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
2120biimpi 208 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
2221breq1d 4898 . . 3 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o))
2319, 22mtbiri 319 . 2 𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
2414, 23pm2.61i 177 1 ¬ {𝐴} ≈ 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1601  wcel 2107  Vcvv 3398  c0 4141  {csn 4398   class class class wbr 4888  suc csuc 5980  ωcom 7345  1oc1o 7838  2oc2o 7839  cen 8240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-om 7346  df-1o 7845  df-2o 7846  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246
This theorem is referenced by:  pmtrsn  18327
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