Theorem en2 7078

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 6996	. . 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 5131	. . . . 5 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
4		dff1o2 5624	. . . . . . . . 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 6675	. . . . . . . 8 ⊢ 2o = {∅, 1o}
7	5, 6	eqtrdi 2283	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→2o → ran 𝑓 = {∅, 1o})
8	7	imaeq2d 5106	. . . . . 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 2281	. . . 4 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → dom 𝑓 = (◡𝑓 “ {∅, 1o}))
11		f1odm 5623	. . . . 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 5632	. . . . . . 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 5620	. . . . . 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 6687	. . . . . 6 ⊢ ∅ ∈ 2o
18	17	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → ∅ ∈ 2o)
19		1lt2o 6688	. . . . . 6 ⊢ 1o ∈ 2o
20	19	a1i 9	. . . . 5 ⊢ ((𝐴 ≈ 2o ∧ 𝑓:𝐴–1-1-onto→2o) → 1o ∈ 2o)
21		fnimapr 5742	. . . . 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 2275	. . 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 5960	. . . . 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 5960	. . . . . 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 3774	. . . . . . 7 ⊢ (𝑦 = (◡𝑓‘1o) → {(◡𝑓‘∅), 𝑦} = {(◡𝑓‘∅), (◡𝑓‘1o)})
30	29	eqeq2d 2246	. . . . . 6 ⊢ (𝑦 = (◡𝑓‘1o) → (𝐴 = {(◡𝑓‘∅), 𝑦} ↔ 𝐴 = {(◡𝑓‘∅), (◡𝑓‘1o)}))
31	30	spcegv 2907	. . . . 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 3773	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥, 𝑦} = {(◡𝑓‘∅), 𝑦})
34	33	eqeq2d 2246	. . . . . 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 2907	. . . 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 1950	1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∅c0 3512  {cpr 3695   class class class wbr 4114  ^◡ccnv 4753  dom cdm 4754  ran crn 4755   “ cima 4757  Fun wfun 5351   Fn wfn 5352  –1-1-onto→wf1o 5356  ‘cfv 5357  1_oc1o 6653  2_oc2o 6654   ≈ cen 6986

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1o 6660  df-2o 6661  df-en 6989

This theorem is referenced by:  en2m  7079  en2prde  7503  upgrex  16224  upgr1een  16245