Theorem 1sdom 8451

Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8314.) (Contributed by Mario Carneiro, 12-Jan-2013.)

Assertion

Ref	Expression
1sdom	⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem 1sdom

Dummy variables 𝑓 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4890	. 2 ⊢ (𝑎 = 𝐴 → (1o ≺ 𝑎 ↔ 1o ≺ 𝐴))
2		rexeq 3331	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3329	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 8003	. . . 4 ⊢ 1o ∈ ω
5		sucdom 8445	. . . 4 ⊢ (1o ∈ ω → (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎)
7		df-2o 7844	. . . 4 ⊢ 2o = suc 1o
8	7	breq1i 4893	. . 3 ⊢ (2o ≼ 𝑎 ↔ suc 1o ≼ 𝑎)
9		2dom 8314	. . . 4 ⊢ (2o ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 7857	. . . . 5 ⊢ 2o = {∅, 1o}
11		vex 3401	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3401	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 5026	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		1oex 7851	. . . . . . . . . . . 12 ⊢ 1o ∈ V
15	11, 12, 13, 14	funpr 6190	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16		df-ne 2970	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 7859	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
18	17	necomi 3023	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1o
19	13, 14, 11, 12	fpr 6687	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21		df-f1 6140	. . . . . . . . . . . . 13 ⊢ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦} ∧ Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
22	20, 21	mpbiran 699	. . . . . . . . . . . 12 ⊢ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ↔ Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
23	13, 11	cnvsn 5873	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 5873	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
25	23, 24	uneq12i 3988	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26		df-pr 4401	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
27	26	cnveqi 5542	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28		cnvun 5792	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
29	27, 28	eqtri 2802	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
30		df-pr 4401	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
31	25, 29, 30	3eqtr4i 2812	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
32	31	funeqi 6156	. . . . . . . . . . . 12 ⊢ (Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
33	22, 32	bitr2i 268	. . . . . . . . . . 11 ⊢ (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
34	15, 16, 33	3imtr3i 283	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
35		prssi 4583	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6356	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎)
37	34, 35, 36	syl2an 589	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎)
38		prex 5141	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39		f1eq1 6346	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎))
40	38, 39	spcev 3502	. . . . . . . . 9 ⊢ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎 → ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
41	37, 40	syl 17	. . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
42		vex 3401	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 8253	. . . . . . . 8 ⊢ ({∅, 1o} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
44	41, 43	sylibr 226	. . . . . . 7 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {∅, 1o} ≼ 𝑎)
45	44	expcom 404	. . . . . 6 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎))
46	45	rexlimivv 3219	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
47	10, 46	syl5eqbr 4921	. . . 4 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → 2o ≼ 𝑎)
48	9, 47	impbii 201	. . 3 ⊢ (2o ≼ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
49	6, 8, 48	3bitr2i 291	. 2 ⊢ (1o ≺ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
50	1, 3, 49	vtoclbg 3468	1 ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1823   ∈ wcel 2107   ≠ wne 2969  ∃wrex 3091   ∪ cun 3790   ⊆ wss 3792  ∅c0 4141  {csn 4398  {cpr 4400  ⟨cop 4404   class class class wbr 4886  ^◡ccnv 5354  suc csuc 5978  Fun wfun 6129  ⟶wf 6131  –1-1→wf1 6132  ωcom 7343  1_oc1o 7836  2_oc2o 7837   ≼ cdom 8239   ≺ csdm 8240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-1o 7843  df-2o 7844  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244

This theorem is referenced by:  unxpdomlem3  8454  frgpnabl  18664  isnzr2  19660