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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1534 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1534.1 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} |
bnj1534.2 | ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) |
Ref | Expression |
---|---|
bnj1534 | ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1534.1 | . 2 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} | |
2 | nfcv 2891 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2891 | . . 3 ⊢ Ⅎ𝑧𝐴 | |
4 | nfv 1909 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≠ (𝐻‘𝑥) | |
5 | bnj1534.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
6 | 5 | nfcii 2879 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
7 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 7 | nffv 6906 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
9 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑥(𝐻‘𝑧) | |
10 | 8, 9 | nfne 3032 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≠ (𝐻‘𝑧) |
11 | fveq2 6896 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
12 | fveq2 6896 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐻‘𝑥) = (𝐻‘𝑧)) | |
13 | 11, 12 | neeq12d 2991 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐻‘𝑥) ↔ (𝐹‘𝑧) ≠ (𝐻‘𝑧))) |
14 | 2, 3, 4, 10, 13 | cbvrabw 3455 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)} |
15 | 1, 14 | eqtri 2753 | 1 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 {crab 3418 ‘cfv 6549 |
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 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 |
This theorem is referenced by: bnj1523 34833 |
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