![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fiunsnnn | GIF version |
Description: Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.) |
Ref | Expression |
---|---|
fiunsnnn | ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∪ {𝐵}) ≈ suc 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 531 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → 𝐴 ≈ 𝑁) | |
2 | en2sn 6840 | . . . 4 ⊢ ((𝐵 ∈ (V ∖ 𝐴) ∧ 𝑁 ∈ ω) → {𝐵} ≈ {𝑁}) | |
3 | 2 | ad2ant2lr 510 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → {𝐵} ≈ {𝑁}) |
4 | simplr 528 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → 𝐵 ∈ (V ∖ 𝐴)) | |
5 | 4 | eldifbd 3156 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → ¬ 𝐵 ∈ 𝐴) |
6 | disjsn 3669 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
7 | 5, 6 | sylibr 134 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∩ {𝐵}) = ∅) |
8 | elirr 4558 | . . . . 5 ⊢ ¬ 𝑁 ∈ 𝑁 | |
9 | disjsn 3669 | . . . . 5 ⊢ ((𝑁 ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ 𝑁) | |
10 | 8, 9 | mpbir 146 | . . . 4 ⊢ (𝑁 ∩ {𝑁}) = ∅ |
11 | 10 | a1i 9 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝑁 ∩ {𝑁}) = ∅) |
12 | unen 6843 | . . 3 ⊢ (((𝐴 ≈ 𝑁 ∧ {𝐵} ≈ {𝑁}) ∧ ((𝐴 ∩ {𝐵}) = ∅ ∧ (𝑁 ∩ {𝑁}) = ∅)) → (𝐴 ∪ {𝐵}) ≈ (𝑁 ∪ {𝑁})) | |
13 | 1, 3, 7, 11, 12 | syl22anc 1250 | . 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∪ {𝐵}) ≈ (𝑁 ∪ {𝑁})) |
14 | df-suc 4389 | . 2 ⊢ suc 𝑁 = (𝑁 ∪ {𝑁}) | |
15 | 13, 14 | breqtrrdi 4060 | 1 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∪ {𝐵}) ≈ suc 𝑁) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∖ cdif 3141 ∪ cun 3142 ∩ cin 3143 ∅c0 3437 {csn 3607 class class class wbr 4018 suc csuc 4383 ωcom 4607 ≈ cen 6765 Fincfn 6767 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-1o 6442 df-er 6560 df-en 6768 |
This theorem is referenced by: php5fin 6911 hashunlem 10819 |
Copyright terms: Public domain | W3C validator |