Theorem snnen2o 9236

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.) Avoid ax-pow 5326, ax-un 7751. (Revised by BTernaryTau, 1-Dec-2024.)

Assertion

Ref	Expression
snnen2o	⊢ ¬ {𝐴} ≈ 2o

Proof of Theorem snnen2o

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df2o3 8484	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5260	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8486	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8495	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 3009	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4819	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1490	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3025	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2957	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1832	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8491	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7290	. . . . . 6 ⊢ ((◡𝑓:2o–1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥})
13	11, 12	mpan2 704	. . . . 5 ⊢ (◡𝑓:2o–1-1→{𝐴} → ∃𝑥2o = {𝑥})
14	10, 13	mto 200	. . . 4 ⊢ ¬ ◡𝑓:2o–1-1→{𝐴}
15		f1ocnv 6837	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6823	. . . . 5 ⊢ (◡𝑓:2o–1-1-onto→{𝐴} → ◡𝑓:2o–1-1→{𝐴})
17	15, 16	syl 18	. . . 4 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1→{𝐴})
18	14, 17	mto 200	. . 3 ⊢ ¬ 𝑓:{𝐴}–1-1-onto→2o
19	18	nex 1832	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5396	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8488	. . 3 ⊢ 2o ∈ V
22		breng 8982	. . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o))
23	20, 21, 22	mp2an 705	. 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)
24	19, 23	mtbir 326	1 ⊢ ¬ {𝐴} ≈ 2o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450  ∅c0 4278  {csn 4583  {cpr 4585   class class class wbr 5102  ^◡ccnv 5649  –1-1→wf1 6535  –1-1-onto→wf1o 6537  1_oc1o 8469  2_oc2o 8470   ≈ cen 8970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-1o 8476  df-2o 8477  df-en 8974

This theorem is used by:  1sdom2  9239  1sdom2dom  9245  pr2ne  10084  pmtrsn  19733  trivnsimpgd  20313