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Theorem exmidsbthr 13207
Description: The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
Assertion
Ref Expression
exmidsbthr (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidsbthr
Dummy variables 𝑖 𝑗 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2144 . . . . 5 (𝑗 = 𝑖 → (𝑗 = ∅ ↔ 𝑖 = ∅))
2 unieq 3740 . . . . . 6 (𝑗 = 𝑖 𝑗 = 𝑖)
32fveq2d 5418 . . . . 5 (𝑗 = 𝑖 → (𝑝 𝑗) = (𝑝 𝑖))
41, 3ifbieq2d 3491 . . . 4 (𝑗 = 𝑖 → if(𝑗 = ∅, 1o, (𝑝 𝑗)) = if(𝑖 = ∅, 1o, (𝑝 𝑖)))
54cbvmptv 4019 . . 3 (𝑗 ∈ ω ↦ if(𝑗 = ∅, 1o, (𝑝 𝑗))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖)))
65mpteq2i 4010 . 2 (𝑝 ∈ ℕ ↦ (𝑗 ∈ ω ↦ if(𝑗 = ∅, 1o, (𝑝 𝑗)))) = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
76exmidsbthrlem 13206 1 (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329   = wceq 1331  c0 3358  ifcif 3469   cuni 3731   class class class wbr 3924  cmpt 3984  EXMIDwem 4113  ωcom 4499  cfv 5118  1oc1o 6299  cen 6625  cdom 6626  xnninf 6998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-exmid 4114  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-1o 6306  df-2o 6307  df-map 6537  df-en 6628  df-dom 6629  df-dju 6916  df-inl 6925  df-inr 6926  df-case 6962  df-omni 6999  df-nninf 7000
This theorem is referenced by:  exmidsbth  13208
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