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| Mirrors > Home > MPE Home > Th. List > Mathboxes > setsnidel | Structured version Visualization version GIF version | ||
| Description: The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.) |
| Ref | Expression |
|---|---|
| setsidel.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
| setsidel.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| setsidel.r | ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) |
| setsnidel.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
| setsnidel.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) |
| setsnidel.s | ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) |
| setsnidel.n | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
| Ref | Expression |
|---|---|
| setsnidel | ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | setsnidel.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
| 2 | 1 | elexd 3504 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ V) |
| 3 | setsnidel.n | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
| 4 | 3 | necomd 2996 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
| 5 | eldifsn 4786 | . . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴)) | |
| 6 | 2, 4, 5 | sylanbrc 583 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴})) |
| 7 | setsnidel.s | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) | |
| 8 | setsnidel.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
| 9 | opelres 6003 | . . . . 5 ⊢ (𝐷 ∈ 𝑌 → (〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ 〈𝐶, 𝐷〉 ∈ 𝑆))) | |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ 〈𝐶, 𝐷〉 ∈ 𝑆))) |
| 11 | 6, 7, 10 | mpbir2and 713 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴}))) |
| 12 | elun1 4182 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ (𝑆 ↾ (V ∖ {𝐴})) → 〈𝐶, 𝐷〉 ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
| 13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
| 14 | setsidel.r | . . 3 ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) | |
| 15 | setsidel.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
| 16 | setsidel.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 17 | setsval 17204 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
| 18 | 15, 16, 17 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
| 19 | 14, 18 | eqtrid 2789 | . 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
| 20 | 13, 19 | eleqtrrd 2844 | 1 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 {csn 4626 〈cop 4632 ↾ cres 5687 (class class class)co 7431 sSet csts 17200 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 |
| This theorem is referenced by: (None) |
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