Theorem 2dom 9070

Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem 2dom

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df2o2 8515	. . . 4 ⊢ 2o = {∅, {∅}}
2	1	breq1i 5150	. . 3 ⊢ (2o ≼ 𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3		brdomi 8999	. . 3 ⊢ ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴)
4	2, 3	sylbi 217	. 2 ⊢ (2o ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴)
5		f1f 6804	. . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → 𝑓:{∅, {∅}}⟶𝐴)
6		0ex 5307	. . . . . 6 ⊢ ∅ ∈ V
7	6	prid1 4762	. . . . 5 ⊢ ∅ ∈ {∅, {∅}}
8		ffvelcdm 7101	. . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
9	5, 7, 8	sylancl 586	. . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘∅) ∈ 𝐴)
10		snex 5436	. . . . . 6 ⊢ {∅} ∈ V
11	10	prid2 4763	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
12		ffvelcdm 7101	. . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
13	5, 11, 12	sylancl 586	. . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14		0nep0 5358	. . . . . 6 ⊢ ∅ ≠ {∅}
15	14	neii 2942	. . . . 5 ⊢ ¬ ∅ = {∅}
16		f1fveq 7282	. . . . . 6 ⊢ ((𝑓:{∅, {∅}}–1-1→𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
17	7, 11, 16	mpanr12 705	. . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
18	15, 17	mtbiri 327	. . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19		eqeq1 2741	. . . . . 6 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
20	19	notbid 318	. . . . 5 ⊢ (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21		eqeq2 2749	. . . . . 6 ⊢ (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
22	21	notbid 318	. . . . 5 ⊢ (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
23	20, 22	rspc2ev 3635	. . . 4 ⊢ (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
24	9, 13, 18, 23	syl3anc 1373	. . 3 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
25	24	exlimiv 1930	. 2 ⊢ (∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
26	4, 25	syl 17	1 ⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070  ∅c0 4333  {csn 4626  {cpr 4628   class class class wbr 5143  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  2_oc2o 8500   ≼ cdom 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-1o 8506  df-2o 8507  df-dom 8987

This theorem is referenced by:  1sdom  9284  1sdomOLD  9285