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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 2932 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2932 | . . 3 ⊢ Ⅎ𝑧𝐴 | |
4 | nfv 1873 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≠ (𝐻‘𝑥) | |
5 | bnj1534.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
6 | 5 | nfcii 2920 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
7 | nfcv 2932 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 7 | nffv 6511 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
9 | nfcv 2932 | . . . 4 ⊢ Ⅎ𝑥(𝐻‘𝑧) | |
10 | 8, 9 | nfne 3070 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≠ (𝐻‘𝑧) |
11 | fveq2 6501 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
12 | fveq2 6501 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐻‘𝑥) = (𝐻‘𝑧)) | |
13 | 11, 12 | neeq12d 3028 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐻‘𝑥) ↔ (𝐹‘𝑧) ≠ (𝐻‘𝑧))) |
14 | 2, 3, 4, 10, 13 | cbvrab 3411 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)} |
15 | 1, 14 | eqtri 2802 | 1 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 {crab 3092 ‘cfv 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-iota 6154 df-fv 6198 |
This theorem is referenced by: bnj1523 31988 |
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