Theorem en2 7065

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 6983	. . 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 5126	. . . . 5 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
4		dff1o2 5619	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→2o ↔ (𝑓 Fn 𝐴 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 2o))
5	4	simp3bi 1041	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = 2o)
6		df2o3 6662	. . . . . . . 8 ⊢ 2o = {∅, 1o}
7	5, 6	eqtrdi 2281	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = {∅, 1o})
8	7	imaeq2d 5101	. . . . . 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 2279	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → dom 𝑓 = (◡𝑓 “ {∅, 1o}))
11		f1odm 5618	. . . . 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 5627	. . . . . . 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 5615	. . . . . 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 6674	. . . . . 6 ⊢ ∅ ∈ 2o
18	17	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ∅ ∈ 2o)
19		1lt2o 6675	. . . . . 6 ⊢ 1o ∈ 2o
20	19	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → 1o ∈ 2o)
21		fnimapr 5737	. . . . 5 ⊢ ((◡𝑓 Fn 2o ∧ ∅ ∈ 2o ∧ 1o ∈ 2o) → (◡𝑓 “ {∅, 1o}) = {(◡𝑓‘∅), (◡𝑓‘1o)})
22	16, 18, 20, 21	syl3anc 1274	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → (◡𝑓 “ {∅, 1o}) = {(◡𝑓‘∅), (◡𝑓‘1o)})
23	10, 12, 22	3eqtr3d 2273	. . 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 5954	. . . . 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 5954	. . . . . 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 3769	. . . . . . 7 ⊢ (𝑦 = (◡𝑓‘1o) → {(◡𝑓‘∅), 𝑦} = {(◡𝑓‘∅), (◡𝑓‘1o)})
30	29	eqeq2d 2244	. . . . . 6 ⊢ (𝑦 = (◡𝑓‘1o) → (𝐴 = {(◡𝑓‘∅), 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)}))
31	30	spcegv 2905	. . . . 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 3768	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥, 𝑦} = {(◡𝑓‘∅), 𝑦})
34	33	eqeq2d 2244	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), 𝑦}))
35	34	exbidv 1874	. . . . 5 ⊢ (𝑥 = (◡𝑓‘∅) → (∃𝑦 𝐴 = {𝑥, 𝑦} ↔ ∃𝑦 𝐴 = {(◡𝑓‘∅), 𝑦}))
36	35	spcegv 2905	. . . 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 1948	1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∅c0 3508  {cpr 3690   class class class wbr 4109  ^◡ccnv 4748  dom cdm 4749  ran crn 4750   “ cima 4752  Fun wfun 5346   Fn wfn 5347  –1-1-onto→wf1o 5351  ‘cfv 5352  1_oc1o 6640  2_oc2o 6641   ≈ cen 6973

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1o 6647  df-2o 6648  df-en 6976

This theorem is referenced by:  en2m  7066  en2prde  7490  upgrex  16098  upgr1een  16119