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Mirrors > Home > MPE Home > Th. List > cda1en | Structured version Visualization version GIF version |
Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
cda1en | ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrefg 7987 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → 𝐴 ≈ 𝐴) |
3 | ensn1g 8021 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) | |
4 | 3 | ensymd 8007 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 1𝑜 ≈ {𝐴}) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → 1𝑜 ≈ {𝐴}) |
6 | simpr 477 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴) | |
7 | disjsn 4246 | . . . 4 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
8 | 6, 7 | sylibr 224 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ∩ {𝐴}) = ∅) |
9 | cdaenun 8996 | . . 3 ⊢ ((𝐴 ≈ 𝐴 ∧ 1𝑜 ≈ {𝐴} ∧ (𝐴 ∩ {𝐴}) = ∅) → (𝐴 +𝑐 1𝑜) ≈ (𝐴 ∪ {𝐴})) | |
10 | 2, 5, 8, 9 | syl3anc 1326 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ (𝐴 ∪ {𝐴})) |
11 | df-suc 5729 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
12 | 10, 11 | syl6breqr 4695 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 {csn 4177 class class class wbr 4653 suc csuc 5725 (class class class)co 6650 1𝑜c1o 7553 ≈ cen 7952 +𝑐 ccda 8989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1o 7560 df-er 7742 df-en 7956 df-cda 8990 |
This theorem is referenced by: pm110.643ALT 9000 pwsdompw 9026 |
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