Theorem snnen2o 8190

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 7764	. . . 4 ⊢ 1𝑜 ∈ ω
2		php5 8189	. . . 4 ⊢ (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ 1𝑜 ≈ suc 1𝑜
4		ensn1g 8062	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
5		df-2o 7606	. . . . . 6 ⊢ 2𝑜 = suc 1𝑜
6	5	eqcomi 2660	. . . . 5 ⊢ suc 1𝑜 = 2𝑜
7	6	breq2i 4693	. . . 4 ⊢ (1𝑜 ≈ suc 1𝑜 ↔ 1𝑜 ≈ 2𝑜)
8		ensymb 8045	. . . . . 6 ⊢ ({𝐴} ≈ 1𝑜 ↔ 1𝑜 ≈ {𝐴})
9		entr 8049	. . . . . . 7 ⊢ ((1𝑜 ≈ {𝐴} ∧ {𝐴} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
10	9	ex 449	. . . . . 6 ⊢ (1𝑜 ≈ {𝐴} → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
11	8, 10	sylbi 207	. . . . 5 ⊢ ({𝐴} ≈ 1𝑜 → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
12	11	con3rr3 151	. . . 4 ⊢ (¬ 1𝑜 ≈ 2𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜))
13	7, 12	sylnbi 319	. . 3 ⊢ (¬ 1𝑜 ≈ suc 1𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜))
14	3, 4, 13	mpsyl 68	. 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
15		2on0 7614	. . . 4 ⊢ 2𝑜 ≠ ∅
16		ensymb 8045	. . . . 5 ⊢ (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
17		en0 8060	. . . . 5 ⊢ (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
18	16, 17	bitri 264	. . . 4 ⊢ (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
19	15, 18	nemtbir 2918	. . 3 ⊢ ¬ ∅ ≈ 2𝑜
20		snprc 4285	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
21	20	biimpi 206	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
22	21	breq1d 4695	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
23	19, 22	mtbiri 316	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
24	14, 23	pm2.61i 176	1 ⊢ ¬ {𝐴} ≈ 2𝑜

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  {csn 4210   class class class wbr 4685  suc csuc 5763  ωcom 7107  1_𝑜c1o 7598  2_𝑜c2o 7599   ≈ cen 7994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000

This theorem is referenced by:  pmtrsn  17985