Theorem snnen2o 8034

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 7606	. . . 4 ⊢ 1𝑜 ∈ ω
2		php5 8033	. . . 4 ⊢ (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ 1𝑜 ≈ suc 1𝑜
4		ensn1g 7907	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
5		df-2o 7448	. . . . . 6 ⊢ 2𝑜 = suc 1𝑜
6	5	eqcomi 2619	. . . . 5 ⊢ suc 1𝑜 = 2𝑜
7	6	breq2i 4591	. . . 4 ⊢ (1𝑜 ≈ suc 1𝑜 ↔ 1𝑜 ≈ 2𝑜)
8		ensymb 7890	. . . . . 6 ⊢ ({𝐴} ≈ 1𝑜 ↔ 1𝑜 ≈ {𝐴})
9		entr 7894	. . . . . . 7 ⊢ ((1𝑜 ≈ {𝐴} ∧ {𝐴} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
10	9	ex 449	. . . . . 6 ⊢ (1𝑜 ≈ {𝐴} → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
11	8, 10	sylbi 206	. . . . 5 ⊢ ({𝐴} ≈ 1𝑜 → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
12	11	con3rr3 150	. . . 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 66	. 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
15		2on0 7456	. . . 4 ⊢ 2𝑜 ≠ ∅
16		ensymb 7890	. . . . 5 ⊢ (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
17		en0 7905	. . . . 5 ⊢ (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
18	16, 17	bitri 263	. . . 4 ⊢ (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
19	15, 18	nemtbir 2877	. . 3 ⊢ ¬ ∅ ≈ 2𝑜
20		snprc 4197	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
21	20	biimpi 205	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
22	21	breq1d 4593	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
23	19, 22	mtbiri 316	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
24	14, 23	pm2.61i 175	1 ⊢ ¬ {𝐴} ≈ 2𝑜

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125   class class class wbr 4583  suc csuc 5642  ωcom 6957  1_𝑜c1o 7440  2_𝑜c2o 7441   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844

This theorem is referenced by:  pmtrsn  17762