Theorem 2dom 9096

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 8523	. . . 4 ⊢ 2o = {∅, {∅}}
2	1	breq1i 5115	. . 3 ⊢ (2o ≼ 𝐴 ↔ {∅, {∅}} ≼ 𝐴)
3		brdomi 9024	. . 3 ⊢ ({∅, {∅}} ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}⟶1-1𝐴)
4	2, 3	sylbi 220	. 2 ⊢ (2o ≼ 𝐴 → ∃𝑓 𝑓:{∅, {∅}}⟶1-1𝐴)
5		f1f 6808	. . . . 5 ⊢ (𝑓:{∅, {∅}}⟶1-1𝐴 → 𝑓:{∅, {∅}}⟶𝐴)
6		0ex 5267	. . . . . 6 ⊢ ∅ ∈ V
7	6	prid1 4725	. . . . 5 ⊢ ∅ ∈ {∅, {∅}}
8		ffvelcdm 7113	. . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ ∅ ∈ {∅, {∅}}) → (𝑓‘∅) ∈ 𝐴)
9	5, 7, 8	sylancl 598	. . . 4 ⊢ (𝑓:{∅, {∅}}⟶1-1𝐴 → (𝑓‘∅) ∈ 𝐴)
10		snex 5404	. . . . . 6 ⊢ {∅} ∈ V
11	10	prid2 4726	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
12		ffvelcdm 7113	. . . . 5 ⊢ ((𝑓:{∅, {∅}}⟶𝐴 ∧ {∅} ∈ {∅, {∅}}) → (𝑓‘{∅}) ∈ 𝐴)
13	5, 11, 12	sylancl 598	. . . 4 ⊢ (𝑓:{∅, {∅}}⟶1-1𝐴 → (𝑓‘{∅}) ∈ 𝐴)
14		0nep0 5326	. . . . . 6 ⊢ ∅ ≠ {∅}
15	14	neii 2961	. . . . 5 ⊢ ¬ ∅ = {∅}
16		f1fveq 7297	. . . . . 6 ⊢ ((𝑓:{∅, {∅}}⟶1-1𝐴 ∧ (∅ ∈ {∅, {∅}} ∧ {∅} ∈ {∅, {∅}})) → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
17	7, 11, 16	mpanr12 718	. . . . 5 ⊢ (𝑓:{∅, {∅}}⟶1-1𝐴 → ((𝑓‘∅) = (𝑓‘{∅}) ↔ ∅ = {∅}))
18	15, 17	mtbiri 330	. . . 4 ⊢ (𝑓:{∅, {∅}}⟶1-1𝐴 → ¬ (𝑓‘∅) = (𝑓‘{∅}))
19		eqeq1 2767	. . . . . 6 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 = 𝑦 ↔ (𝑓‘∅) = 𝑦))
20	19	notbid 321	. . . . 5 ⊢ (𝑥 = (𝑓‘∅) → (¬ 𝑥 = 𝑦 ↔ ¬ (𝑓‘∅) = 𝑦))
21		eqeq2 2775	. . . . . 6 ⊢ (𝑦 = (𝑓‘{∅}) → ((𝑓‘∅) = 𝑦 ↔ (𝑓‘∅) = (𝑓‘{∅})))
22	21	notbid 321	. . . . 5 ⊢ (𝑦 = (𝑓‘{∅}) → (¬ (𝑓‘∅) = 𝑦 ↔ ¬ (𝑓‘∅) = (𝑓‘{∅})))
23	20, 22	rspc2ev 3592	. . . 4 ⊢ (((𝑓‘∅) ∈ 𝐴 ∧ (𝑓‘{∅}) ∈ 𝐴 ∧ ¬ (𝑓‘∅) = (𝑓‘{∅})) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
24	9, 13, 18, 23	syl3anc 1399	. . 3 ⊢ (𝑓:{∅, {∅}}⟶1-1𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
25	24	exlimiv 1965	. 2 ⊢ (∃𝑓 𝑓:{∅, {∅}}⟶1-1𝐴 → ∃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 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  ∅c0 4281  {csn 4588  {cpr 4590   class class class wbr 5108  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566  2_oc2o 8508   ≼ cdom 9009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fv 6574  df-1o 8514  df-2o 8515  df-dom 9013

This theorem is used by:  1sdom  9284