Theorem exmidsbthr 16915

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	⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID)

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidsbthr

Dummy variables 𝑖 𝑗 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2241	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑗 = ∅ ↔ 𝑖 = ∅))
2		unieq 3928	. . . . . 6 ⊢ (𝑗 = 𝑖 → ∪ 𝑗 = ∪ 𝑖)
3	2	fveq2d 5679	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑝‘∪ 𝑗) = (𝑝‘∪ 𝑖))
4	1, 3	ifbieq2d 3651	. . . 4 ⊢ (𝑗 = 𝑖 → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))
5	4	cbvmptv 4211	. . 3 ⊢ (𝑗 ∈ ω ↦ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))
6	5	mpteq2i 4202	. 2 ⊢ (𝑝 ∈ ℕ∞ ↦ (𝑗 ∈ ω ↦ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))) = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))
7	6	exmidsbthrlem 16914	1 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398  ∅c0 3512  ifcif 3624  ∪ cuni 3919   class class class wbr 4114   ↦ cmpt 4176  EXMIDwem 4312  ωcom 4717  ‘cfv 5357  1_oc1o 6653   ≈ cen 6986   ≼ cdom 6987  ℕ_∞xnninf 7423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-exmid 4313  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-map 6897  df-en 6989  df-dom 6990  df-dju 7342  df-inl 7351  df-inr 7352  df-case 7388  df-nninf 7424  df-omni 7439

This theorem is referenced by:  exmidsbth  16916