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Theorem en2eleq 8714
 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 7607 . . . . . 6 2𝑜 ∈ ω
2 nnfi 8038 . . . . . 6 (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
31, 2ax-mp 5 . . . . 5 2𝑜 ∈ Fin
4 enfi 8061 . . . . 5 (𝑃 ≈ 2𝑜 → (𝑃 ∈ Fin ↔ 2𝑜 ∈ Fin))
53, 4mpbiri 247 . . . 4 (𝑃 ≈ 2𝑜𝑃 ∈ Fin)
65adantl 481 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ∈ Fin)
7 simpl 472 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋𝑃)
8 1onn 7606 . . . . . . . . 9 1𝑜 ∈ ω
98a1i 11 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10 simpr 476 . . . . . . . . 9 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11 df-2o 7448 . . . . . . . . 9 2𝑜 = suc 1𝑜
1210, 11syl6breq 4624 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13 dif1en 8078 . . . . . . . 8 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑋𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
149, 12, 7, 13syl3anc 1318 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15 en1uniel 7914 . . . . . . 7 ((𝑃 ∖ {𝑋}) ≈ 1𝑜 (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
1614, 15syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
17 eldifsn 4260 . . . . . 6 ( (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ ( (𝑃 ∖ {𝑋}) ∈ 𝑃 (𝑃 ∖ {𝑋}) ≠ 𝑋))
1816, 17sylib 207 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → ( (𝑃 ∖ {𝑋}) ∈ 𝑃 (𝑃 ∖ {𝑋}) ≠ 𝑋))
1918simpld 474 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ 𝑃)
20 prssi 4293 . . . 4 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃)
217, 19, 20syl2anc 691 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃)
2218simprd 478 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
2322necomd 2837 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋 (𝑃 ∖ {𝑋}))
24 pr2nelem 8710 . . . . 5 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃𝑋 (𝑃 ∖ {𝑋})) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
257, 19, 23, 24syl3anc 1318 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
26 ensym 7891 . . . . 5 (𝑃 ≈ 2𝑜 → 2𝑜𝑃)
2726adantl 481 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 2𝑜𝑃)
28 entr 7894 . . . 4 (({𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜 ∧ 2𝑜𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
2925, 27, 28syl2anc 691 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
30 fisseneq 8056 . . 3 ((𝑃 ∈ Fin ∧ {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
316, 21, 29, 30syl3anc 1318 . 2 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
3231eqcomd 2616 1 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  ∪ cuni 4372   class class class wbr 4583  suc csuc 5642  ωcom 6957  1𝑜c1o 7440  2𝑜c2o 7441   ≈ cen 7838  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845 This theorem is referenced by:  en2other2  8715  psgnunilem1  17736
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