Theorem en2 6911

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 6834	. . 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 5045	. . . . 5 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
4		dff1o2 5526	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→2o ↔ (𝑓 Fn 𝐴 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 2o))
5	4	simp3bi 1016	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = 2o)
6		df2o3 6515	. . . . . . . 8 ⊢ 2o = {∅, 1o}
7	5, 6	eqtrdi 2253	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = {∅, 1o})
8	7	imaeq2d 5021	. . . . . 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 2251	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → dom 𝑓 = (◡𝑓 “ {∅, 1o}))
11		f1odm 5525	. . . . 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 5534	. . . . . . 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 5522	. . . . . 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 6526	. . . . . 6 ⊢ ∅ ∈ 2o
18	17	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ∅ ∈ 2o)
19		1lt2o 6527	. . . . . 6 ⊢ 1o ∈ 2o
20	19	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → 1o ∈ 2o)
21		fnimapr 5638	. . . . 5 ⊢ ((◡𝑓 Fn 2o ∧ ∅ ∈ 2o ∧ 1o ∈ 2o) → (◡𝑓 “ {∅, 1o}) = {(◡𝑓‘∅), (◡𝑓‘1o)})
22	16, 18, 20, 21	syl3anc 1249	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (◡𝑓 “ {∅, 1o}) = {(◡𝑓‘∅), (◡𝑓‘1o)})
23	10, 12, 22	3eqtr3d 2245	. . 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 5849	. . . . 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 5849	. . . . . 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 3710	. . . . . . 7 ⊢ (𝑦 = (◡𝑓‘1o) → {(◡𝑓‘∅), 𝑦} = {(◡𝑓‘∅), (◡𝑓‘1o)})
30	29	eqeq2d 2216	. . . . . 6 ⊢ (𝑦 = (◡𝑓‘1o) → (𝐴 = {(◡𝑓‘∅), 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)}))
31	30	spcegv 2860	. . . . 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 3709	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥, 𝑦} = {(◡𝑓‘∅), 𝑦})
34	33	eqeq2d 2216	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), 𝑦}))
35	34	exbidv 1847	. . . . 5 ⊢ (𝑥 = (◡𝑓‘∅) → (∃𝑦 𝐴 = {𝑥, 𝑦} ↔ ∃𝑦 𝐴 = {(◡𝑓‘∅), 𝑦}))
36	35	spcegv 2860	. . . 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 1921	1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∅c0 3459  {cpr 3633   class class class wbr 4043  ^◡ccnv 4673  dom cdm 4674  ran crn 4675   “ cima 4677  Fun wfun 5264   Fn wfn 5265  –1-1-onto→wf1o 5269  ‘cfv 5270  1_oc1o 6494  2_oc2o 6495   ≈ cen 6824

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1o 6501  df-2o 6502  df-en 6827

This theorem is referenced by: (None)