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Mirrors > Home > MPE Home > Th. List > Mathboxes > nna1iscard | Structured version Visualization version GIF version |
Description: For any natural number, the add one operation is results in a cardinal number. (Contributed by RP, 1-Oct-2023.) |
Ref | Expression |
---|---|
nna1iscard | ⊢ (𝑁 ∈ ω → (𝑁 +o 1o) ∈ ran card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7858 | . . . 4 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On) | |
2 | oa1suc 8532 | . . . 4 ⊢ (𝑁 ∈ On → (𝑁 +o 1o) = suc 𝑁) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑁 ∈ ω → (𝑁 +o 1o) = suc 𝑁) |
4 | peano2 7878 | . . 3 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω) | |
5 | 3, 4 | jca 511 | . 2 ⊢ (𝑁 ∈ ω → ((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω)) |
6 | simpl 482 | . . 3 ⊢ (((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω) → (𝑁 +o 1o) = suc 𝑁) | |
7 | simpr 484 | . . 3 ⊢ (((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω) → suc 𝑁 ∈ ω) | |
8 | 6, 7 | eqeltrd 2827 | . 2 ⊢ (((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω) → (𝑁 +o 1o) ∈ ω) |
9 | omssrncard 42872 | . . 3 ⊢ ω ⊆ ran card | |
10 | 9 | sseli 3973 | . 2 ⊢ ((𝑁 +o 1o) ∈ ω → (𝑁 +o 1o) ∈ ran card) |
11 | 5, 8, 10 | 3syl 18 | 1 ⊢ (𝑁 ∈ ω → (𝑁 +o 1o) ∈ ran card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ran crn 5670 Oncon0 6358 suc csuc 6360 (class class class)co 7405 ωcom 7852 1oc1o 8460 +o coa 8464 cardccrd 9932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 |
This theorem is referenced by: (None) |
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