Theorem en2 6993

Description: A set equinumerous to ordinal 2 is an unordered pair. (Contributed by Mario Carneiro, 5-Jan-2016.)

Assertion

Ref	Expression
en2	⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem en2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6912	. . 3 ⊢ (𝐴 ≈ 2o ↔ ∃𝑓 𝑓:𝐴–1-1-onto→2o)
2	1	biimpi 120	. 2 ⊢ (𝐴 ≈ 2o → ∃𝑓 𝑓:𝐴–1-1-onto→2o)
3		cnvimarndm 5098	. . . . 5 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
4		dff1o2 5585	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→2o ↔ (𝑓 Fn 𝐴 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 2o))
5	4	simp3bi 1038	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = 2o)
6		df2o3 6592	. . . . . . . 8 ⊢ 2o = {∅, 1o}
7	5, 6	eqtrdi 2278	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = {∅, 1o})
8	7	imaeq2d 5074	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→2o → (◡𝑓 “ ran 𝑓) = (◡𝑓 “ {∅, 1o}))
9	8	adantl 277	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (◡𝑓 “ ran 𝑓) = (◡𝑓 “ {∅, 1o}))
10	3, 9	eqtr3id 2276	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → dom 𝑓 = (◡𝑓 “ {∅, 1o}))
11		f1odm 5584	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→2o → dom 𝑓 = 𝐴)
12	11	adantl 277	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → dom 𝑓 = 𝐴)
13		f1ocnv 5593	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→2o → ◡𝑓:2o–1-1-onto→𝐴)
14	13	adantl 277	. . . . . 6 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ◡𝑓:2o–1-1-onto→𝐴)
15		f1ofn 5581	. . . . . 6 ⊢ (◡𝑓:2o–1-1-onto→𝐴 → ◡𝑓 Fn 2o)
16	14, 15	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ◡𝑓 Fn 2o)
17		0lt2o 6604	. . . . . 6 ⊢ ∅ ∈ 2o
18	17	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ∅ ∈ 2o)
19		1lt2o 6605	. . . . . 6 ⊢ 1o ∈ 2o
20	19	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → 1o ∈ 2o)
21		fnimapr 5702	. . . . 5 ⊢ ((◡𝑓 Fn 2o ∧ ∅ ∈ 2o ∧ 1o ∈ 2o) → (◡𝑓 “ {∅, 1o}) = {(◡𝑓‘∅), (◡𝑓‘1o)})
22	16, 18, 20, 21	syl3anc 1271	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (◡𝑓 “ {∅, 1o}) = {(◡𝑓‘∅), (◡𝑓‘1o)})
23	10, 12, 22	3eqtr3d 2270	. . 3 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → 𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)})
24		simpr 110	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → 𝑓:𝐴–1-1-onto→2o)
25		f1ocnvdm 5917	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→2o ∧ ∅ ∈ 2o) → (◡𝑓‘∅) ∈ 𝐴)
26	24, 17, 25	sylancl 413	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (◡𝑓‘∅) ∈ 𝐴)
27		f1ocnvdm 5917	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→2o ∧ 1o ∈ 2o) → (◡𝑓‘1o) ∈ 𝐴)
28	24, 19, 27	sylancl 413	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (◡𝑓‘1o) ∈ 𝐴)
29		preq2 3747	. . . . . . 7 ⊢ (𝑦 = (◡𝑓‘1o) → {(◡𝑓‘∅), 𝑦} = {(◡𝑓‘∅), (◡𝑓‘1o)})
30	29	eqeq2d 2241	. . . . . 6 ⊢ (𝑦 = (◡𝑓‘1o) → (𝐴 = {(◡𝑓‘∅), 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)}))
31	30	spcegv 2892	. . . . 5 ⊢ ((◡𝑓‘1o) ∈ 𝐴 → (𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)} → ∃𝑦 𝐴 = {(◡𝑓‘∅), 𝑦}))
32	28, 31	syl 14	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)} → ∃𝑦 𝐴 = {(◡𝑓‘∅), 𝑦}))
33		preq1 3746	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥, 𝑦} = {(◡𝑓‘∅), 𝑦})
34	33	eqeq2d 2241	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), 𝑦}))
35	34	exbidv 1871	. . . . 5 ⊢ (𝑥 = (◡𝑓‘∅) → (∃𝑦 𝐴 = {𝑥, 𝑦} ↔ ∃𝑦 𝐴 = {(◡𝑓‘∅), 𝑦}))
36	35	spcegv 2892	. . . 4 ⊢ ((◡𝑓‘∅) ∈ 𝐴 → (∃𝑦 𝐴 = {(◡𝑓‘∅), 𝑦} → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}))
37	26, 32, 36	sylsyld 58	. . 3 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)} → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}))
38	23, 37	mpd 13	. 2 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})
39	2, 38	exlimddv 1945	1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∅c0 3492  {cpr 3668   class class class wbr 4086  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   “ cima 4726  Fun wfun 5318   Fn wfn 5319  –1-1-onto→wf1o 5323  ‘cfv 5324  1_oc1o 6570  2_oc2o 6571   ≈ cen 6902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-2o 6578  df-en 6905

This theorem is referenced by:  en2m  6994  en2prde  7389  upgrex  15944