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Mirrors > Home > MPE Home > Th. List > Mathboxes > iotaexeu | Structured version Visualization version GIF version |
Description: The iota class exists. This theorem does not require ax-nul 4717 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotaexeu | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval 5779 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
2 | 1 | eqcomd 2616 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
3 | 2 | eximi 1752 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑)) |
4 | df-eu 2462 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
5 | isset 3180 | . 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 280 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 Vcvv 3173 ℩cio 5766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-sbc 3403 df-un 3545 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 |
This theorem is referenced by: iotasbc 37642 pm14.18 37651 iotavalb 37653 sbiota1 37657 |
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