Theorem 2dom 9063

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 8490	. . . 4 ⊢ 2o = {∅, {∅}}
2	1	breq1i 5109	. . 3 ⊢ (2o ≼ 𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3		brdomi 8991	. . 3 ⊢ ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴)
4	2, 3	sylbi 220	. 2 ⊢ (2o ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴)
5		f1f 6781	. . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → 𝑓:{∅, {∅}}⟶𝐴)
6		0ex 5260	. . . . . 6 ⊢ ∅ ∈ V
7	6	prid1 4719	. . . . 5 ⊢ ∅ ∈ {∅, {∅}}
8		ffvelcdm 7084	. . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
9	5, 7, 8	sylancl 598	. . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘∅) ∈ 𝐴)
10		snex 5396	. . . . . 6 ⊢ {∅} ∈ V
11	10	prid2 4720	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
12		ffvelcdm 7084	. . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
13	5, 11, 12	sylancl 598	. . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14		0nep0 5318	. . . . . 6 ⊢ ∅ ≠ {∅}
15	14	neii 2957	. . . . 5 ⊢ ¬ ∅ = {∅}
16		f1fveq 7268	. . . . . 6 ⊢ ((𝑓:{∅, {∅}}–1-1→𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
17	7, 11, 16	mpanr12 718	. . . . 5 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
18	15, 17	mtbiri 330	. . . 4 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19		eqeq1 2764	. . . . . 6 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
20	19	notbid 321	. . . . 5 ⊢ (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21		eqeq2 2772	. . . . . 6 ⊢ (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
22	21	notbid 321	. . . . 5 ⊢ (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
23	20, 22	rspc2ev 3588	. . . 4 ⊢ (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
24	9, 13, 18, 23	syl3anc 1398	. . 3 ⊢ (𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
25	24	exlimiv 1962	. 2 ⊢ (∃𝑓 𝑓:{∅, {∅}}–1-1→𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
26	4, 25	syl 18	1 ⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  ∅c0 4277  {csn 4584  {cpr 4586   class class class wbr 5102  ⟶wf 6537  –1-1→wf1 6538  ‘cfv 6541  2_oc2o 8475   ≼ cdom 8976

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 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  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 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fv 6549  df-1o 8481  df-2o 8482  df-dom 8980

This theorem is used by:  1sdom  9251