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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 7809 | . . . 4 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On) | |
2 | oa1suc 8478 | . . . 4 ⊢ (𝑁 ∈ On → (𝑁 +o 1o) = suc 𝑁) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑁 ∈ ω → (𝑁 +o 1o) = suc 𝑁) |
4 | peano2 7828 | . . 3 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω) | |
5 | 3, 4 | jca 513 | . 2 ⊢ (𝑁 ∈ ω → ((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω)) |
6 | simpl 484 | . . 3 ⊢ (((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω) → (𝑁 +o 1o) = suc 𝑁) | |
7 | simpr 486 | . . 3 ⊢ (((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω) → suc 𝑁 ∈ ω) | |
8 | 6, 7 | eqeltrd 2838 | . 2 ⊢ (((𝑁 +o 1o) = suc 𝑁 ∧ suc 𝑁 ∈ ω) → (𝑁 +o 1o) ∈ ω) |
9 | omssrncard 41819 | . . 3 ⊢ ω ⊆ ran card | |
10 | 9 | sseli 3941 | . 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 397 = wceq 1542 ∈ wcel 2107 ran crn 5635 Oncon0 6318 suc csuc 6320 (class class class)co 7358 ωcom 7803 1oc1o 8406 +o coa 8410 cardccrd 9872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9876 |
This theorem is referenced by: (None) |
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