Theorem snnen2o 9201

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 5336, ax-un 7732. (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 8457	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5270	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8459	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8468	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 3012	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4825	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1490	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3028	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2960	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1830	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8464	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7280	. . . . . 6 ⊢ ((◡𝑓:2o–1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥})
13	11, 12	mpan2 703	. . . . 5 ⊢ (◡𝑓:2o–1-1→{𝐴} → ∃𝑥2o = {𝑥})
14	10, 13	mto 200	. . . 4 ⊢ ¬ ◡𝑓:2o–1-1→{𝐴}
15		f1ocnv 6833	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6819	. . . . 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 1830	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5410	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8461	. . 3 ⊢ 2o ∈ V
22		breng 8948	. . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o))
23	20, 21, 22	mp2an 704	. 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)
24	19, 23	mtbir 326	1 ⊢ ¬ {𝐴} ≈ 2o

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286  {csn 4589  {cpr 4591   class class class wbr 5109  ^◡ccnv 5660  –1-1→wf1 6533  –1-1-onto→wf1o 6535  1_oc1o 8442  2_oc2o 8443   ≈ cen 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8449  df-2o 8450  df-en 8940

This theorem is referenced by:  1sdom2  9204  1sdom2dom  9210  pr2ne  9985  pmtrsn  19584  trivnsimpgd  20164