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Mirrors > Home > MPE Home > Th. List > dju1en | 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 |
---|---|
dju1en | ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrefg 8538 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → 𝐴 ≈ 𝐴) |
3 | ensn1g 8571 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
4 | 3 | ensymd 8557 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ≈ {𝐴}) |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → 1o ≈ {𝐴}) |
6 | simpr 487 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴) | |
7 | disjsn 4644 | . . . 4 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
8 | 6, 7 | sylibr 236 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ∩ {𝐴}) = ∅) |
9 | djuenun 9593 | . . 3 ⊢ ((𝐴 ≈ 𝐴 ∧ 1o ≈ {𝐴} ∧ (𝐴 ∩ {𝐴}) = ∅) → (𝐴 ⊔ 1o) ≈ (𝐴 ∪ {𝐴})) | |
10 | 2, 5, 8, 9 | syl3anc 1366 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ (𝐴 ∪ {𝐴})) |
11 | df-suc 6194 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
12 | 10, 11 | breqtrrdi 5105 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 {csn 4564 class class class wbr 5063 suc csuc 6190 1oc1o 8092 ≈ cen 8503 ⊔ cdju 9324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-ord 6191 df-on 6192 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-1st 7686 df-2nd 7687 df-1o 8099 df-er 8286 df-en 8507 df-dju 9327 |
This theorem is referenced by: dju1p1e2ALT 9597 pwsdompw 9623 |
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