Theorem 2dom 3332

Description: A set that dominates ordinal 2 has at least 2 different members.

Hypothesis

Ref	Expression
2dom.1	⊢ A ∈ V

Assertion

Ref	Expression
2dom	⊢ (2o ≼ A → ∃x ∈ A ∃y ∈ A ¬ x = y)

Distinct variable group(s):   x,y,A

Proof of Theorem 2dom

Step	Hyp	Ref	Expression
1		df2o2 3112	. . . 4 ⊢ 2o = {∅, {∅}}
2	1	breq1i 2068	. . 3 ⊢ (2o ≼ A ↔ {∅, {∅}} ≼ A)
3		2dom.1	. . . 4 ⊢ A ∈ V
4	3	brdom 3283	. . 3 ⊢ ({∅, {∅}} ≼ A ↔ ∃f f:{∅, {∅}}–1-1→A)
5	2, 4	bitr 151	. 2 ⊢ (2o ≼ A ↔ ∃f f:{∅, {∅}}–1-1→A)
6		cleq1 1107	. . . . . 6 ⊢ (x = (f ‘∅) → (x = y ↔ (f ‘∅) = y))
7	6	negbid 463	. . . . 5 ⊢ (x = (f ‘∅) → (¬ x = y ↔ ¬ (f ‘∅) = y))
8		cleq2 1110	. . . . . 6 ⊢ (y = (f ‘{∅}) → ((f ‘∅) = y ↔ (f ‘∅) = (f ‘{∅})))
9	8	negbid 463	. . . . 5 ⊢ (y = (f ‘{∅}) → (¬ (f ‘∅) = y ↔ ¬ (f ‘∅) = (f ‘{∅})))
10	7, 9	rcla42ev 1405	. . . 4 ⊢ ((((f ‘∅) ∈ A ∧ (f ‘{∅}) ∈ A) ∧ ¬ (f ‘∅) = (f ‘{∅})) → ∃x ∈ A ∃y ∈ A ¬ x = y)
11		f1f 2781	. . . . 5 ⊢ (f:{∅, {∅}}–1-1→A → f:{∅, {∅}}–→A)
12		0ex 1745	. . . . . . . 8 ⊢ ∅ ∈ V
13	12	pri1 1841	. . . . . . 7 ⊢ ∅ ∈ {∅, {∅}}
14		ffvrn 2890	. . . . . . 7 ⊢ ((f:{∅, {∅}}–→A ∧ ∅ ∈ {∅, {∅}}) → (f ‘∅) ∈ A)
15	13, 14	mpan2 519	. . . . . 6 ⊢ (f:{∅, {∅}}–→A → (f ‘∅) ∈ A)
16		p0ex 1885	. . . . . . . 8 ⊢ {∅} ∈ V
17	16	pri2 1842	. . . . . . 7 ⊢ {∅} ∈ {∅, {∅}}
18		ffvrn 2890	. . . . . . 7 ⊢ ((f:{∅, {∅}}–→A ∧ {∅} ∈ {∅, {∅}}) → (f ‘{∅}) ∈ A)
19	17, 18	mpan2 519	. . . . . 6 ⊢ (f:{∅, {∅}}–→A → (f ‘{∅}) ∈ A)
20	15, 19	jca 236	. . . . 5 ⊢ (f:{∅, {∅}}–→A → ((f ‘∅) ∈ A ∧ (f ‘{∅}) ∈ A))
21	11, 20	syl 12	. . . 4 ⊢ (f:{∅, {∅}}–1-1→A → ((f ‘∅) ∈ A ∧ (f ‘{∅}) ∈ A))
22		0nep0 1887	. . . . 5 ⊢ ¬ ∅ = {∅}
23	13, 17	pm3.2i 234	. . . . . 6 ⊢ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})
24		f1fveq 2918	. . . . . 6 ⊢ ((f:{∅, {∅}}–1-1→A ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((f ‘∅) = (f ‘{∅}) ↔ ∅ = {∅}))
25	23, 24	mpan2 519	. . . . 5 ⊢ (f:{∅, {∅}}–1-1→A → ((f ‘∅) = (f ‘{∅}) ↔ ∅ = {∅}))
26	22, 25	mtbiri 539	. . . 4 ⊢ (f:{∅, {∅}}–1-1→A → ¬ (f ‘∅) = (f ‘{∅}))
27	10, 21, 26	sylanc 361	. . 3 ⊢ (f:{∅, {∅}}–1-1→A → ∃x ∈ A ∃y ∈ A ¬ x = y)
28	27	19.23aiv 952	. 2 ⊢ (∃f f:{∅, {∅}}–1-1→A → ∃x ∈ A ∃y ∈ A ¬ x = y)
29	5, 28	sylbi 174	1 ⊢ (2o ≼ A → ∃x ∈ A ∃y ∈ A ¬ x = y)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  ∅c0 1707  {csn 1808  {cpr 1809   class class class wbr 2054  –→wf 2418  –1-1→wf1 2419   ‘cfv 2422  2_oc2o 3100   ≼ cdom 3272

This theorem is referenced by:  unxpdomlem 3649

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-1o 3104  df-2o 3105  df-dom 3275

metamath.org