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Mirrors > Home > ILE Home > Th. List > Mathboxes > exmidsbthr | GIF version |
Description: The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.) |
Ref | Expression |
---|---|
exmidsbthr | ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2184 | . . . . 5 ⊢ (𝑗 = 𝑖 → (𝑗 = ∅ ↔ 𝑖 = ∅)) | |
2 | unieq 3818 | . . . . . 6 ⊢ (𝑗 = 𝑖 → ∪ 𝑗 = ∪ 𝑖) | |
3 | 2 | fveq2d 5519 | . . . . 5 ⊢ (𝑗 = 𝑖 → (𝑝‘∪ 𝑗) = (𝑝‘∪ 𝑖)) |
4 | 1, 3 | ifbieq2d 3558 | . . . 4 ⊢ (𝑗 = 𝑖 → if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)) = if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) |
5 | 4 | cbvmptv 4099 | . . 3 ⊢ (𝑗 ∈ ω ↦ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) |
6 | 5 | mpteq2i 4090 | . 2 ⊢ (𝑝 ∈ ℕ∞ ↦ (𝑗 ∈ ω ↦ if(𝑗 = ∅, 1o, (𝑝‘∪ 𝑗)))) = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))) |
7 | 6 | exmidsbthrlem 14652 | 1 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1351 = wceq 1353 ∅c0 3422 ifcif 3534 ∪ cuni 3809 class class class wbr 4003 ↦ cmpt 4064 EXMIDwem 4194 ωcom 4589 ‘cfv 5216 1oc1o 6409 ≈ cen 6737 ≼ cdom 6738 ℕ∞xnninf 7117 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-exmid 4195 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-1o 6416 df-2o 6417 df-map 6649 df-en 6740 df-dom 6741 df-dju 7036 df-inl 7045 df-inr 7046 df-case 7082 df-nninf 7118 df-omni 7132 |
This theorem is referenced by: exmidsbth 14654 |
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