Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > invdisjrabw | Structured version Visualization version GIF version |
Description: Version of invdisjrab 5045 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
invdisjrabw | ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2976 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
2 | nfcv 2976 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcsb1v 3900 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 | |
4 | 3 | nfeq1 2992 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 = 𝑦 |
5 | csbeq1a 3890 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶) | |
6 | 5 | eqeq1d 2822 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶 = 𝑦 ↔ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
7 | 1, 2, 4, 6 | elrabf 3672 | . . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) |
8 | simprr 771 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦) | |
9 | 7, 8 | sylan2b 595 | . . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦) |
10 | 9 | rgen2 3202 | . 2 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 |
11 | invdisj 5043 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) | |
12 | 10, 11 | ax-mp 5 | 1 ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 {crab 3141 ⦋csb 3876 Disj wdisj 5024 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-disj 5025 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |