Theorem snnen2o 8417

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 7985	. . . 4 ⊢ 1o ∈ ω
2		php5 8416	. . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ 1o ≈ suc 1o
4		ensn1g 8286	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
5		df-2o 7826	. . . . . 6 ⊢ 2o = suc 1o
6	5	eqcomi 2833	. . . . 5 ⊢ suc 1o = 2o
7	6	breq2i 4880	. . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o)
8		ensymb 8269	. . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴})
9		entr 8273	. . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o)
10	9	ex 403	. . . . . 6 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o))
11	8, 10	sylbi 209	. . . . 5 ⊢ ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o))
12	11	con3rr3 153	. . . 4 ⊢ (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
13	7, 12	sylnbi 322	. . 3 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o))
14	3, 4, 13	mpsyl 68	. 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
15		2on0 7835	. . . 4 ⊢ 2o ≠ ∅
16		ensymb 8269	. . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
17		en0 8284	. . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
18	16, 17	bitri 267	. . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
19	15, 18	nemtbir 3093	. . 3 ⊢ ¬ ∅ ≈ 2o
20		snprc 4470	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
21	20	biimpi 208	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
22	21	breq1d 4882	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o))
23	19, 22	mtbiri 319	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o)
24	14, 23	pm2.61i 177	1 ⊢ ¬ {𝐴} ≈ 2o

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3413  ∅c0 4143  {csn 4396   class class class wbr 4872  suc csuc 5964  ωcom 7325  1_oc1o 7818  2_oc2o 7819   ≈ cen 8218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-om 7326  df-1o 7825  df-2o 7826  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224

This theorem is referenced by:  pmtrsn  18289