Theorem snnen2o 8149

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 7719	. . . 4 ⊢ 1𝑜 ∈ ω
2		php5 8148	. . . 4 ⊢ (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ 1𝑜 ≈ suc 1𝑜
4		ensn1g 8021	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
5		df-2o 7561	. . . . . 6 ⊢ 2𝑜 = suc 1𝑜
6	5	eqcomi 2631	. . . . 5 ⊢ suc 1𝑜 = 2𝑜
7	6	breq2i 4661	. . . 4 ⊢ (1𝑜 ≈ suc 1𝑜 ↔ 1𝑜 ≈ 2𝑜)
8		ensymb 8004	. . . . . 6 ⊢ ({𝐴} ≈ 1𝑜 ↔ 1𝑜 ≈ {𝐴})
9		entr 8008	. . . . . . 7 ⊢ ((1𝑜 ≈ {𝐴} ∧ {𝐴} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
10	9	ex 450	. . . . . 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 320	. . 3 ⊢ (¬ 1𝑜 ≈ suc 1𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜))
14	3, 4, 13	mpsyl 68	. 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
15		2on0 7569	. . . 4 ⊢ 2𝑜 ≠ ∅
16		ensymb 8004	. . . . 5 ⊢ (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
17		en0 8019	. . . . 5 ⊢ (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
18	16, 17	bitri 264	. . . 4 ⊢ (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
19	15, 18	nemtbir 2889	. . 3 ⊢ ¬ ∅ ≈ 2𝑜
20		snprc 4253	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
21	20	biimpi 206	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
22	21	breq1d 4663	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
23	19, 22	mtbiri 317	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
24	14, 23	pm2.61i 176	1 ⊢ ¬ {𝐴} ≈ 2𝑜

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177   class class class wbr 4653  suc csuc 5725  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  pmtrsn  17939