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Theorem en2eleq 8776
Description: Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Assertion
Ref Expression
en2eleq ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})

Proof of Theorem en2eleq
StepHypRef Expression
1 2onn 7666 . . . . . 6 2𝑜 ∈ ω
2 nnfi 8098 . . . . . 6 (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
31, 2ax-mp 5 . . . . 5 2𝑜 ∈ Fin
4 enfi 8121 . . . . 5 (𝑃 ≈ 2𝑜 → (𝑃 ∈ Fin ↔ 2𝑜 ∈ Fin))
53, 4mpbiri 248 . . . 4 (𝑃 ≈ 2𝑜𝑃 ∈ Fin)
65adantl 482 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ∈ Fin)
7 simpl 473 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋𝑃)
8 1onn 7665 . . . . . . . . 9 1𝑜 ∈ ω
98a1i 11 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10 simpr 477 . . . . . . . . 9 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11 df-2o 7507 . . . . . . . . 9 2𝑜 = suc 1𝑜
1210, 11syl6breq 4659 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13 dif1en 8138 . . . . . . . 8 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑋𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
149, 12, 7, 13syl3anc 1323 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15 en1uniel 7973 . . . . . . 7 ((𝑃 ∖ {𝑋}) ≈ 1𝑜 (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
1614, 15syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
17 eldifsn 4292 . . . . . 6 ( (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ ( (𝑃 ∖ {𝑋}) ∈ 𝑃 (𝑃 ∖ {𝑋}) ≠ 𝑋))
1816, 17sylib 208 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → ( (𝑃 ∖ {𝑋}) ∈ 𝑃 (𝑃 ∖ {𝑋}) ≠ 𝑋))
1918simpld 475 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ 𝑃)
20 prssi 4326 . . . 4 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃)
217, 19, 20syl2anc 692 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃)
2218simprd 479 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
2322necomd 2851 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋 (𝑃 ∖ {𝑋}))
24 pr2nelem 8772 . . . . 5 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃𝑋 (𝑃 ∖ {𝑋})) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
257, 19, 23, 24syl3anc 1323 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
26 ensym 7950 . . . . 5 (𝑃 ≈ 2𝑜 → 2𝑜𝑃)
2726adantl 482 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 2𝑜𝑃)
28 entr 7953 . . . 4 (({𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜 ∧ 2𝑜𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
2925, 27, 28syl2anc 692 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
30 fisseneq 8116 . . 3 ((𝑃 ∈ Fin ∧ {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
316, 21, 29, 30syl3anc 1323 . 2 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
3231eqcomd 2632 1 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wne 2796  cdif 3557  wss 3560  {csn 4153  {cpr 4155   cuni 4407   class class class wbr 4618  suc csuc 5687  ωcom 7013  1𝑜c1o 7499  2𝑜c2o 7500  cen 7897  Fincfn 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-2o 7507  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904
This theorem is referenced by:  en2other2  8777  psgnunilem1  17829
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