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Theorem en2eleq 9013
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 7881 . . . . . 6 2𝑜 ∈ ω
2 nnfi 8310 . . . . . 6 (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
31, 2ax-mp 5 . . . . 5 2𝑜 ∈ Fin
4 enfi 8333 . . . . 5 (𝑃 ≈ 2𝑜 → (𝑃 ∈ Fin ↔ 2𝑜 ∈ Fin))
53, 4mpbiri 248 . . . 4 (𝑃 ≈ 2𝑜𝑃 ∈ Fin)
65adantl 473 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ∈ Fin)
7 simpl 474 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋𝑃)
8 1onn 7880 . . . . . . . . 9 1𝑜 ∈ ω
98a1i 11 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10 simpr 479 . . . . . . . . 9 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11 df-2o 7722 . . . . . . . . 9 2𝑜 = suc 1𝑜
1210, 11syl6breq 4837 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13 dif1en 8350 . . . . . . . 8 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑋𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
149, 12, 7, 13syl3anc 1473 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15 en1uniel 8185 . . . . . . 7 ((𝑃 ∖ {𝑋}) ≈ 1𝑜 (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
1614, 15syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
17 eldifsn 4454 . . . . . 6 ( (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ ( (𝑃 ∖ {𝑋}) ∈ 𝑃 (𝑃 ∖ {𝑋}) ≠ 𝑋))
1816, 17sylib 208 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → ( (𝑃 ∖ {𝑋}) ∈ 𝑃 (𝑃 ∖ {𝑋}) ≠ 𝑋))
1918simpld 477 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ 𝑃)
20 prssi 4490 . . . 4 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃)
217, 19, 20syl2anc 696 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃)
2218simprd 482 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
2322necomd 2979 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋 (𝑃 ∖ {𝑋}))
24 pr2nelem 9009 . . . . 5 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃𝑋 (𝑃 ∖ {𝑋})) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
257, 19, 23, 24syl3anc 1473 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
26 ensym 8162 . . . . 5 (𝑃 ≈ 2𝑜 → 2𝑜𝑃)
2726adantl 473 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 2𝑜𝑃)
28 entr 8165 . . . 4 (({𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜 ∧ 2𝑜𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
2925, 27, 28syl2anc 696 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
30 fisseneq 8328 . . 3 ((𝑃 ∈ Fin ∧ {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
316, 21, 29, 30syl3anc 1473 . 2 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
3231eqcomd 2758 1 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wne 2924  cdif 3704  wss 3707  {csn 4313  {cpr 4315   cuni 4580   class class class wbr 4796  suc csuc 5878  ωcom 7222  1𝑜c1o 7714  2𝑜c2o 7715  cen 8110  Fincfn 8113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-1o 7721  df-2o 7722  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117
This theorem is referenced by:  en2other2  9014  psgnunilem1  18105
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