MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en2eleq Structured version   Visualization version   GIF version

Theorem en2eleq 9021
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 7889 . . . . . 6 2𝑜 ∈ ω
2 nnfi 8318 . . . . . 6 (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
31, 2ax-mp 5 . . . . 5 2𝑜 ∈ Fin
4 enfi 8341 . . . . 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 7888 . . . . . . . . 9 1𝑜 ∈ ω
98a1i 11 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10 simpr 479 . . . . . . . . 9 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11 df-2o 7730 . . . . . . . . 9 2𝑜 = suc 1𝑜
1210, 11syl6breq 4845 . . . . . . . 8 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13 dif1en 8358 . . . . . . . 8 ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜𝑋𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
149, 12, 7, 13syl3anc 1477 . . . . . . 7 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15 en1uniel 8193 . . . . . . 7 ((𝑃 ∖ {𝑋}) ≈ 1𝑜 (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
1614, 15syl 17 . . . . . 6 ((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
17 eldifsn 4462 . . . . . 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 4498 . . . 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 2987 . . . . 5 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑋 (𝑃 ∖ {𝑋}))
24 pr2nelem 9017 . . . . 5 ((𝑋𝑃 (𝑃 ∖ {𝑋}) ∈ 𝑃𝑋 (𝑃 ∖ {𝑋})) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
257, 19, 23, 24syl3anc 1477 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜)
26 ensym 8170 . . . . 5 (𝑃 ≈ 2𝑜 → 2𝑜𝑃)
2726adantl 473 . . . 4 ((𝑋𝑃𝑃 ≈ 2𝑜) → 2𝑜𝑃)
28 entr 8173 . . . 4 (({𝑋, (𝑃 ∖ {𝑋})} ≈ 2𝑜 ∧ 2𝑜𝑃) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
2925, 27, 28syl2anc 696 . . 3 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃)
30 fisseneq 8336 . . 3 ((𝑃 ∈ Fin ∧ {𝑋, (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
316, 21, 29, 30syl3anc 1477 . 2 ((𝑋𝑃𝑃 ≈ 2𝑜) → {𝑋, (𝑃 ∖ {𝑋})} = 𝑃)
3231eqcomd 2766 1 ((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  cdif 3712  wss 3715  {csn 4321  {cpr 4323   cuni 4588   class class class wbr 4804  suc csuc 5886  ωcom 7230  1𝑜c1o 7722  2𝑜c2o 7723  cen 8118  Fincfn 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-2o 7730  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125
This theorem is referenced by:  en2other2  9022  psgnunilem1  18113
  Copyright terms: Public domain W3C validator