Theorem snnen2o 8707

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

Step	Hyp	Ref	Expression
1		1onn 8265	. . . 4 ⊢ 1o ∈ ω
2		php5 8705	. . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ 1o ≈ suc 1o
4		ensn1g 8574	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
5		df-2o 8103	. . . . . 6 ⊢ 2o = suc 1o
6	5	eqcomi 2830	. . . . 5 ⊢ suc 1o = 2o
7	6	breq2i 5074	. . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o)
8		ensymb 8557	. . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴})
9		entr 8561	. . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o)
10	9	ex 415	. . . . . 6 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o))
11	8, 10	sylbi 219	. . . . 5 ⊢ ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o))
12	11	con3rr3 158	. . . 4 ⊢ (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
13	7, 12	sylnbi 332	. . 3 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
14	3, 4, 13	mpsyl 68	. 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
15		2on0 8113	. . . 4 ⊢ 2o ≠ ∅
16		ensymb 8557	. . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
17		en0 8572	. . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
18	16, 17	bitri 277	. . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
19	15, 18	nemtbir 3112	. . 3 ⊢ ¬ ∅ ≈ 2o
20		snprc 4653	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
21	20	biimpi 218	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
22	21	breq1d 5076	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o))
23	19, 22	mtbiri 329	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
24	14, 23	pm2.61i 184	1 ⊢ ¬ {𝐴} ≈ 2o

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  {csn 4567   class class class wbr 5066  suc csuc 6193  ωcom 7580  1_oc1o 8095  2_oc2o 8096   ≈ cen 8506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-1o 8102  df-2o 8103  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512

This theorem is referenced by:  pmtrsn  18647  trivnsimpgd  19219